Actually the DMSO will not cause damage for molecules of the extract. The plant materials completely extracted using DMSO will not be useful for any other analysis methods, so it is better to extract with the different polar solvents and dry it and mix with DMSO to the right concentration. This could help to save the plant extract and use it for further analysis.
DMSO not any role for the stability. It is use for the proper dissolving the substance those are not dissolve in the organic and inorganic solvent.Your plant extract is dissolve in the particular solvent than no required the dissolve in the DMSO. You directly use extract for the bacterial inhibition purpose.
Once extract are in liquid form they are not that much stable as in solid powders. Thats why we always use freshly prepared solution for any activity. DMSO is generally used to dissolve the plants extract when other solvents fails to do the same.