You probably do want to keep water out as freezing and thawing could
damage asphalt. I'd probably consider asphalt crack sealer if not that
deep or maybe a roofing cement chalking. You also want to seal the
asphalt if you haven't already. I got a new drive a couple of years ago
and watched it absorb water. Installer sealed it after 6 months and I
thought coating was skimpy and repeated it myself this year.