Call up your Pro Tools session and open up the Edit window. Find your drum tracks and start with the kick drum. Highlight the entire track, click on "Windows" in the file menu and select "Show Strip Silence."
Keep the defaults shown in Strip Silence and slowly move "Strip Threshold" to the right until you see boxes forming around each kick drum region. Be sure to move it slowly, because you want to isolate the drums but you don't want to cut off the tail completely. When you've finished this, click the button marked "Strip."
Repeat step two for the snare, toms and hi-hats, but don't bother with the crash or ride cymbals. These cymbals require a long tail or else they'll sound like a retro drum machine.
When finished, go back through all the tracks and add short fade-ins and fade-outs to each region to smooth out the sound. Listen carefully to each track to hear if you need to cut off the tails even more.
When you've finished cleaning the drum tracks, highlight the regions of each drum track and choose "Consolidate Selection" under "Edit" in the file menu. This will consolidate all the regions into one continuous track and make it easier to organize when it comes time to mix tracks.
Add a room reverb to the finished drum tracks to emulate a live room sound. If you really do want that '80s sound, be sure to leave the reverb tail very short and include a predelay of 20 milliseconds.