HIDs use significantly less power than halogen bulbs... There should be less current draw with the new lights. I would look for a problem with the installation. Maybe one of the ballast power wires is shorting to ground.
The stock alternator outputs something like 50-60 amps... 160 would be serious overkill. '92-95 Civics use a slightly more powerful alternator (an extra 10 amps or so), I would look into that. The newer alts use a belt that's 1 rib wider than the CRX belt, so you would need to swap over either the crank pulley from the Civic, or your old alternator pulley onto the new alt. Otherwise it should be a bolt in, plug 'n play swap.