I don't think it is DOSBox, because the sound works perfectly in SC2, but I can't get it to work in SC1. Any ideas?

I always thought the PC version of SC1 doesn't have digital sounds (Amiga version does). There's plenty of choice for noise by MIDI instruments, though:
(command line parameters)

/S:mt32 Roland MT-32 or LAPC sound and music (SC1 Only)
/S:adlib Adlib sound and music
/S:cms CMS sound and music (SC1 Only)
/S:tandy Tandy 3-voice sound and music (SC1 Only)
/S:internal Internal speaker sound and music
/S:silent No sound and music

So I suggest checking if you have MIDI working in DOSbox.

You're missing the sound switch.
Thankfully the devs at Gog were kind enough to provide the manual
"C:\Program Files (x86)\GOG Galaxy\Games\Star Control\manual.pdf"


With notepad open:
"C:\Program Files (x86)\GOG Galaxy\Games\Star Control\dosboxSC1_single.conf"

Add this switch
/S:adlib

For example

[autoexec]
# Lines in this section will be run at startup.
mount C ".."
c:
cls
starcon.exe /S:adlib
exit