My husband is here on a student visa (F-1). He has a social #, but no work authorization. Do I claim him on my taxes?

He doesn't have any source of income.

You can file married filing jointly.  This would be completed by preparing your return in TurboTax and printing it out, attaching a 6013 g election statement you prepare outside of TurboTax. and mailing the return to the IRS,

Please see attached link for IRS Publication 519, page 9, Publication 519 US Tax Guide for Aliens 

Please see link for treating your nonresident spouse as a resident.https://www.irs.gov/individuals/international-taxpayers/nonresident-alien-spouse

You will both need to include your worldwide income on the return.  Please see page 9 of the Publication 519 for the information you need to end this election choice.

Ok, but can I file separately? He doesn't have any income, not even from another country. He's been living in U.S. since 2014.

Yes, you can file separately however it is almost always better to file married filing jointly.  If you choose to file separately There's only one specific case where you can claim your spouse's personal exemption:

    You're filing your return as Married Filing Separately; and
    Your spouse had no income; and
    Your spouse isn't filing a return; and
    Your spouse isn't being claimed by somebody else as a dependent.

All of the 3 applies to him (is there still a point to file jointly?). If I claim him on my taxes, do we still need to fill out the form 6013g?

No.  If you file married filing separately, no election is required. You can just claim him on your married filing separately resident return.  I only brought up MFJ election as your tax rates will be higher on a MFS return.