This is not necessarily a good thing, since some juries now think that DNA evidence is required for a guilty verdict, leading to hung juries and such, but TV crime shows have definitely increased the demand for and belief in DNA evidence. Whether or not DNA evidence is required for most verdicts, there is increased interest in it.
Yes, TV crime shows have increased the demand for DNA evidence, because the people think that there should be DNA evidence in each and every case. Jurors also think that DNA evidence is the only true evidence; they think that everything else is circumstantial. It is much harder for the District Attorney.
The contribution of crime shows to DNA collection is likely the raising of awareness of its existence as a type of valuable form of evidence. Crime shows do not contribute to its demand, however. Only the application of it's use in a setting that demonstrates its efficacy in the criminal justice system will increase its demand.
I do not believe TV crime shows have increased demand for DNA evidence. These shows do little to mirror the actually practice of evidence gathering and law enforcement jobs. They simply aren't realistic and they haven't done much to change procedures in the real field. Plenty of cases are tried without any real evidence or DNA samples.