Would it be possible to equip an asteroid to collect space junk in Earth's orbit?

This is plausible, and might even be a good idea if used right.

First off, NASA has been working on plans for an asteroid redirect mission, called ARM. While it remains to be seen whether this will be approved, and they plan to put it into lunar orbit, this is a hypothetical scenario anyways so I'll ignore that.

Now, putting an asteroid into earth orbit is a little bit of a difficult subject. Sure, you could do it, but messing up would net you a whole lot of problems. I presume it would have to go into high earth orbit if it goes into orbit at all.

This rules out using the asteroid directly like a satellite, but that would have been impractical even if it was sanctioned by everyone with enough influence to have an opinion on earth. From there the satellite would be best used as a base of operations, from which you could send drones to collect junk and keep refueling stations. I'd also include a telescope or two, but that's personal preference.

If you want a list of targets though, here would be a good place to look.