The potential impact of close passes is significant to astronomers, who have already been looking at these events, past and future. Close passing stars are ominously called nemesis stars, since they could cause comet showers, extinction events, and even planetary ejection from a stellar system. It's also interesting from the standpoint of exchange of material between star systems. We know for certain this happens, because the material returned by the Stardust mission from the Wild-2 comet has a different nitrogen isotope ratio then the rest of the solar system - the comet originated from a different star. It's likely this occurred because another star captured a comet with a hyperbolic orbit after it escaped its parent star's gravity, but could this occur by "direct" capture?
You can see from the figure above that there are no close shaves in our future, assuming our surveys have detected all nearby objects* and our proper and radial motion calculations are correct. But there is one coming nearby: in 31,000 years, the Sun's neighbor Epsilon Eridani will be less than 1 LY to Luyten 726-8AB, close enough to disturb a hypothetical Oort cloud around the system, even if not the dust disk around the star; the nemesis will be closer than 1 LY for 4,6000 years. (One of the interesting things about these encounters is that they happen on a scale of thousands of years, pretty fast in comparison to most astronomical events.)
So what are the closest encounters likely to have happened to our neighbors? This paper by Deltorn and Kalas first looked for encounters with Vega, Fomalhaut, and Epsilon Eridani over the past million years, because they have asymmetric debris disks. They found that these stars respectively had 4, 6, and 3 encounters of (at closest) 1.6 LY (no word on how that frequency of encounters compares to other stars with less strange debris disks or no disk at all).
In the course of looking at Hipparcos data for 21,497 nearby stars, they did find an extremely close encounter 350,000 years ago between Alpha Fornacis (HD 20010) and HD 17848, at 0.265 LY. For reference, that's 16521 AU. The comet with the farthest aphelion known is a NEAP-discovered object C/2002_L9, with an aphelion of 14245 AU. There are surely more, so the encounter between the two stars above is surely close enough to capture and therefore exchange material.
The reason this is interesting is that if life exists elsewhere in the universe, and it can move between stellar systems, we're most likely to detect artifacts on low gravity bodies**, especially if they're self-replicating. Comets and small icy moons in our own solar system are a good place to start. Whether what we find are von Neumann probes or space algae is unimportant, and in any event likely indecipherable at first discovery. The same arguments certainly apply to any objects moving under their own power. If these objects are going to move to another system, they're more likely to do it when the two systems are close. This makes a recent finding by Forgan, Papadogiannakis, and Kitching all the more interesting: that using gravity-assist is faster than powered flight alone by two orders of magnitude for galactic exploration, and even adding power to gravity-assist doesn't appreciably speed it along. Their conclusion is that the velocities achieved in their model are still slower than they would need to be in order to say "no aliens have come to the solar system, and if they were around, they would have by now, so there aren't any." (There's more below about the first part of that statement.) This is all to say, if gravity-assist is really the way to go, then it's possible for there to be interstellar "backwaters", where you can't get there from here.
Powered and gravity-assist models of interstellar travel.
In the gravity-assist model there can be "backwaters"
that are very difficult to get to
If aliens are traveling through the galaxy by powered flight, there's less sensitivity to the distribution of stars, and no sensitivity to starting point: you can turn in any direction you want and travel straight there, and one direction is as good as another. But in a gravity-assist model, there are constrained paths from one star to another. That is, over a few thousand years, you might be able to get to one 20 LY away but you can't get to the one 3 LY away because of your starting point, and the lack of stellar companions or other bodies you might need for your slingshot.
So if it's not just distance, but also the detailed distribution of interstellar geography that matters to the spread of replicator material between star systems (either probes or "living things"), periods of anomalously close contact between stars make a difference. Previous I argued that, to increase our chances of detecting life in other stellar systems, we should look at systems with super-Earths (more gravity therefore thicker atmosphere to preserve simultaneous gas and liquid phases, and more surface area for replicator chemistry to use as reaction vessels); we should also look toward the galactic core - the stars there are older (have had more time for life to evolve) and they're distributed more densely. In addition we should focus on stars that have had close encounters like this, as spreading zones. If it's possible for life to move between systems, it's most likely to move between close systems. Pejoratively, think of it like this: promiscuous stars are the ones most likely to catch something!
So is there anything interesting about the two close-encounter stars? HD 20010 is an F-class star, fairly near at 46 LY from Earth. The system has an IR excess, which means dust. The star is 2.9 billion years old, and is just leaving the main sequence. For comparison, at the corresponding point in the solar system's history, eukaryotes were only just starting to pull ahead of bacteria, but the ancestors of flagellates and ciliates hadn't yet split, and the atmosphere was still only 3% oxygen. Trilobites were still far-future science fiction at that point. The other star HD17848 is now 165 LY from Earth; it's an A-class which also has an infrared excess and therefore dust disk.
Regarding the original three stars in this paper, it turns out Fomalhaut does have a very strange dust ring and a solar system with Earth-like planets.
At this point, strangely dusty planetary systems that have recently been perturbed by near-misses from nemesis stars are best explained by what we already know in astronomy: that there were multiple impact events and the dust is either settling or some non-miraculous process is replenishing the disk. That said, it's still worth paying special attention to "promiscuous" stellar systems for anomalous findings.
*It's risky to assume we've discovered all "nearby" objects. The second and third closest known stars to the solar system were respectively discovered less than a century ago, and this year.
**Detecting extrasolar life by detecting their signals carries far more assumptions than looking for artifacts; for e.g., that they're intelligent, that they use similar technology, that they don't mind being overheard or for some reason want to be overheard by other intelligences, and that, most dubiously, we notice and realize what we're listening to when we detect one. Material artifacts are more likely to be recognizable against background and give the added benefit of proof that transit by replicators between stars is possible.