There's a lot more to it than that; in fact, I'd say the DNA on the shawl is among the weakest evidence in favor.
If you're up for a good read, I'd recommend this book.
Here are a few highlights:
Scotland Yard thought it was Kosminksi.
There was one eye witness in the case who got a good look at the killer. That man positively identified Kosminksi. However, he refused to testify once he discovered that Kosminksi was Jewish. Without that, they couldn't prosecute him.
Nevertheless, Scotland Yard had him followed day and night. That coincided exactly with the end of the murders.
Based on similar sorts of murders committed since, the murderer was most likely a paranoid schizophrenic -- a rare mental disorder. Kosminski showed symptoms consistent with paranoid schizophrenia (hearing voices, extreme grandiosity, irrational fear of being poisoned, etc.).
The murders were sexual in nature. Kosminski had problems with compulsive masturbation.
After being essentially confined to his house, Kosminski attacked his sister with a knife (all the murders were committed against women, using a knife). He was taken away to an insane asylum, where he stayed for the rest of his life, as his paranoia worsened.
In general, modern psychological profiling points to someone very much like Kosminksi.
Geographic profiling (predicting the living/working location of the killer based on the location of the murders) also perfectly matches Kosminksi's known residences and places of work.
There's more to it, and there are a lot of gaps and contradictions in the historical record which make it impossible to say with total certainty who the killer was, but the balance of the evidence points pretty convincingly to Kosminksi.