My husband & I filed "Married Filing Sep." since 2013. He owes taxes for 2015/16. If we file jointly in 2018 do I become responsible for these past taxes?

Are you in a Community Property State?

Colorado

Not a community property state

No, you do not become responsible for his past tax obligations.  You are only responsible for tax obligations for tax years in which you filed Married filing jointly.  

However, if you are due a refund in 2018, the refund will be seized to pay his past tax obligations unless you file for injured spouse relief.

The rules are different in the nine community property states. These jurisdictions view all marital property as owned equally by both spouses, including wages and earnings.

Colorado, not a community property state.