It seems that my computer has a SCSI-1 adapter board (adaptec
   1542C) and a SCSI-2 hard drive (seagate ST3600N).

The 1542C can talk to SCSI 2 disks just fine.

   DOS, [...] have no problems identifying the geometry of my hard
   drive, but NetBSD seems to have problems doing so.  Is there any
   way around this?

It's impossible for me to help you unless you specify what the
symptoms of the `problems' are, and what version of NetBSD you're
trying to boot.