Can I claim my husband as a dependent if he lives in Mexico?

A spouse can  NEVER be  claimed as a dependent ... a married couple simply files a joint return no matter where they live or how much they make.

If you are a US citizen and your spouse does not have a Social Security number or an ITIN and you are not applying for an ITIN with the tax return then you can only file your tax return as Married Filing Separately.  

Where you are asked to enter the spouse's Social Security number enter 999-88-9999.  You can only print and mail your tax return; it cannot be e-filed.  When you print the tax return erase the Social Security number for your spouse and manually enter NRA for non-resident alien.

See this TurboTax support FAQ for the procedure to print and mail a tax return using the online editions - https://ttlc.intuit.com/questions/1944348-how-do-i-print-and-mail-my-return-in-turbotax-online

If you are filing your tax return as Married Filing Jointly you must apply for an ITIN with the tax return and you would need to be providing a statement with your tax return that you want your Nonresident Alien Spouse Treated as a Resident.  See IRS Publication 54 Tax Guide for U.S. Citizens and Resident Aliens Abroad pages 6 and 7 - https://www.irs.gov/pub/irs-pdf/p54.pdf

Go to this IRS website for ITIN information - https://www.irs.gov/individuals/international-taxpayers/general-itin-information

No, a spouse is never a dependent.  If you are legally married you have to file as either Married Filing Jointly or Married Filing Separately.  

To file as Married Filing Jointly your spouse must have a valid Social Security number that can be used for employment purposes or an ITIN issued to them by the IRS.