I like compulsory voting, it seems to work fairly well in countries like
Australia. Many people probably do care but just need the extra push to go and vote. And for those who really don't care, you can make the penalty small enough to not really affect people. Or show up and abstain/spoil the ballot/write in a candidate who won't win in protest. Regardless, it does help combat extremists turning up in disproportionately large numbers to sway elections results, and encourages people to inform themselves about the politics.
If we don't do compulsory voting, there are several other ways to increase voter turnout, most of which is just making voting easier to do so people who couldn't or didn't want to put in the effort to vote now can (edit: and should probably be used in conjunction with compulsory voting if that is implemented):
1. Make voting by mail easier to do. California has an excellent Permanent Vote by Mail status that you can sign up for, without providing any reason, just by ticking a box at voter registration (which is online, so if you forget or need to change the mailing address you can just fill the form out again). Additionally, all ballots postmarked by Election Day and received within 3 days are counted (stamp is included, and you can track your ballot's status online). It's definitely a model I think other places should follow.
2. Make voting on a weekend and/or a public holiday. Furthermore/if that fails, increase worker protections and time allowed for people to vote.
3. Make understanding the election more accessible: have better and more informative sample ballots/official election guides, translate into more languages, etc.
Edit: forgot 4. Make registering to vote easier. Allow registering to vote online, registering day of election (and voting conditionally upon approval of application), potentially relax/remove voter ID laws (in the US only, since the US has no officially assigned ID that everyone has).
Further edit: Actually, completely forgot about automatic voter registration, I think that's probably the first option for making registering to vote easier.