If back taxes are owed on a joint return, then the tax liability is owned equally by both of you. When filing a tax return under Married Filing Separate, both names and Social Security numbers are entered on both forms for IRS reference.
If you were still married as of December 31, 2017, you must file Married Filing Joint, Married Filing Separate or in some cases Head of Household.
The following Intuit links may help you determine which filing status is best for you:
https://ttlc.intuit.com/questions/1894449-is-it-better-for-a-married-couple-to-file-jointly-or-separ... (MFJ vs. MFS)
https://ttlc.intuit.com/questions/1894553-do-i-qualify-for-head-of-household?jump_to=answer_3288637 (HOH)
However, if the back taxes were due to one spouse's income and you file Married Filing Joint, the other spouse may file Form 8379, Injured Spouse Allocation. This Intuit link shows you how and gives additional information
https://ttlc.intuit.com/questions/1910698-how-do-i-file-form-8379-injured-spouse-allocation
