The use of DNA testing has been hotly debated in a number of cases, e.g., proving parenting to obtain child support, checking if a person died to permit his wife to remarry, preventing inheritance, etc.
Poskim are not in unanimous agreement but generally not inclined to use DNA testing, e.g., because of lack of 100% reliability (with the grave consequences of e.g., a woman being married twice) or second-order issues (e.g., creating mamzerim), although in certain specific situations DNA tests have been used to strengthen other proofs.
R Shlomo Brody (Yeshivat Hakotel) writes about the mix of opinions
Dayan Shlomo Dichovsky argued that DNA testing cannot provide absolute accuracy to deem a child a mamzer, but is sufficient to
create enough doubt to prevent inheritance or child support (Assia 5).
Other colleagues, including Rabbi Avraham Shapira, rejected any distinctions and all DNA evidence.
Rabbi Shmuel Vosner has also distinguished the admissibility of DNA evidence in different areas of law, stating that it remains insufficient to convict criminals or create mamzerim, but can be used
in inheritance cases when there are no competing claims, and in
certain cases of agunot (Tehumin 21).
Rabbi Shlomo Auerbach allowed DNA evidence to identify babies mistakenly confused in a hospital, and further asserted that it could
be used for paternity testing if its science becomes universally
accepted (Nishmat Avraham EH 4:35).
Following this assertion, Rabbi Zalman Goldberg and the Rabbinical Council of America's judicial court employed DNA (and other) evidence
to prevent spouses of 9/11 victims from becoming agunot.
While some continue to contend that we can still ignore DNA evidence to prevent mamzerim, others believe that when confronted
with incontrovertible scientific evidence, Jewish law must
recognize the truth, no matter which way it cuts.
I asked the CEO of the leading Israeli genealogy software company who told me that DNA tests could be used with 98% reliability to tell someone he was Jewish. This might be good to encourage someone to convert but might not be a halachic proof.
So it seems that, for now, and possibly waiting for better DNA testing technology, DNA testing cannot be the sole evidence of Jewishness. As Monica Cellio wrote in the comments above, the solution in case of presumption of Jewishness is conversion, which makes it an easier problem to solve than the issues discussed above.
For further reading, see here, here, here and there.