It depends almost entirely on the circumstances. If they are holding a doorway or stairway, somewhere where they can only be attacked pretty much one-on-one, they could probably hold off an almost indefinite number, at least until either exhaustion set in or the attackers' morale collapsed. But if sent out into a large mob that can surround them... they'd probably kill a couple but it would only take a relatively small advantage in numbers to overwhelm them, grab them from behind, and so on. On horseback they stand a better chance but even then they're not invincible and would have to keep moving. If they get bogged down they can be pulled out of the saddle and killed.
In the King's Landing riots we saw how even otherwise fairly decent and well-equipped warriors were overwhelmed by numbers, on horseback or no. They mght have been the absolute cream of the crop but in that sort of situation skill is not as important as it might be.
I'm going to say "maybe", but it depends on a lot going right for Viserys. He'll need to get the Dothraki to Westeros, which is tricky as discussed, but if he manages to land them in a reasonably coordinated fashion he's in business. Let's say that works out and he manages to assemble enough of a fleet to do it while fending off Stannis, say the Redwynes and Velaryons. I'll also assume he brings the Golden Company, which gives them some backbone, and it was the GC's plan to join him, after all. And let's say Dorne rises for Viserys too, as was Doran's intention. For the sake of argument, let's also say that the candidate for kingship isn't Viserys himself but Aegon, and that the Dothraki for some reason agree to this.
The addition of Dorne and the GC gives Viserys a substantial boost in size, possibly up to double his original numbers, and a big enough army to look threatening. The key is getting the Reach on side, and that's going to be tricky but not necessarily impossible. The Reach were after all Targ loyalists the first time round, unlike any of the northern kingdoms. Viserys and Dany are both already tied up marriage-wise, but Aegon isn't, and he might be an appropriate match for Margaery especially if he were the claimant king. (Doran isn't going to mind either way on the Viserys/Aegon issue, as either Arianne becomes queen or Elia's son becomes king).
The next trick is to target the potential neutrals and at worst guarantee their neutrality, and at best turn them into allies. The Vale should not be too difficult to persuade to stay out of things if they're listening to Lysa, but if Jaime's appointment has gone through he ironically might be able to raise the knights of the Vale more successfully, in his role as Warden. The Iron Islands could perhaps be bought by an offer to relinquish the Targ claim over them and grant them independence, and encourage them to raid the Westerlands and the North. Although not a full kingdom, the Freys are also important allies to seek, as they can hold that crossing almost indefinitely and stop the Northern armies linking up with the Westerlands and Riverlands in the west, forcing them down towards King's Landing.
Assuming they get the Reach onside somehow, though, there's then the question of where they go next. Renly demonstrated that King's Landing is vulnerable from the south, but Renly didn't have to fight his way through the Stormlands. The best option might be to send the bulk of the Reach troops to deal with the Stormlands and block the Roseroad to starve King's Landing (as Renly did), while the Dothraki and GC descend on the Riverlands and do to them what Tywin did. It will take the North a while to mobilise, longer if the Iron Islands are raiding their coast, and knocking out the southern kingdoms before the North can get there will be key. Assuming Tywin responds directly and brings his army into the Riverlands, they'll want to beat him in a pitched battle and try to force him back beyond the Golden Tooth. If not the Dothraki can run rings around Tywin's army in the Riverlands as Robb did.
But that's a lot of ifs. Even in that scenario the whole thing could be buggered by the Vale, who if they entered the war could link up with the eastern North army, and then move on to defend King's Landing to the point of impregnability, catch the Riverlands contingent in a pincer with Tywin, or reinforce the Stormlords against the Reach.
They don't know what we know. To them the Night's Watch are a weird anachronism that's useful for disposing of criminals and the rest of the time they're happy to ignore. Wights, Others, giants, and the like are fairy tales.
Let's turn it round. What do they know, what have they heard, that should make them care? And there's a question of priority, too: even if they do care about some mythical monsters wandering about north of the Wall, their neighbour's trying to burn their winter supplies is probably more of an immediate concern.