The strategies used against Cultist and Marauder, the two offensive Ruins-giving cards, are similar to the strategies used against Curse-givers in other sets. Ruins are not quite as bad as Curses, since the cards have a little bit of utility and don't carry a VP penalty.
That said: Dark Ages has lots of light trashing to help you handle both unwanted Coppers and Shelters, and any Ruins that might come your way. Hermit is a dynamite Ruins-trasher, since you don't actually have to line it up in the same hand and it can give you Silver (or more Hermits) to overcome deck dilution. Junk Dealer cleans your deck up and provides a bit of buying power too. Forager doesn't usually give much buying power, but it's cheap.
A little more obliquely, Procession can give you a benefit and clear out the Ruins as well, although you'll want other targets for it. Death Cart inflicts Ruins on yourself--but $5 is an incredible chunk of change, and you won't lack for targets if your opponents are giving you Ruins as well. If you have lots of junk, Sage can help you wade through the bad stuff to find your Junk Dealers or Altars. Count can clear a lot of junk from your deck at once; put the only good card on top of your deck, then choose the "trash your hand" option. And Catacombs deals with junk by letting you discard it and draw something else.
You brought up Beggar and having two Silvers is pretty great. The gain-three-Coppers option, if you don't get attacked, depends a lot on the deck. It's great for Duchy and alternative-VP rushes (Gardens, of course...); less so if you have Provinces in mind for the endgame.
One last note: Rats looks like an attractive trasher, and it is powerful in that regard, but then you have to get rid of the Rats! I think Rats combines best with trash-for-benefit cards like Apprentice, Bishop, and Salvager, and not so well with most Dark Ages cards.