I will mute the unwanted strings with the fingers that are doing the fretting. If it is the sixth string, my thumb will sometimes do the muting and sometimes the finger fretting the fifth string.
For a D7 - X5453X, I'll probably use my ring finger to mute the 6th and my index to mute the 1st.
For a D7 - X00212, I'll likely use my thumb (or sometimes fret the 6th string at the 2nd fret with my thumb.)