Short answer: No.
In November of 2015, the CDC issued a warning about rising STI/STD rates. Many news outlets reported this with bold and scary headlines. Further, several outlets erroneously and wrongly attributed this rise in STI rates to PrEP, a link not made by the CDC nor supported by the available evidence.
What did the CDC actually say? The largest jump in new STI cases in the USA were in 15-24 year old men and women, mainly for gonorrhea and chlamydia. Interviews with CDC spokespeople outlined that much of this rise was attributable to significant drops in funding for STI / STD education programs at health departments across the country. They went on to say that economic and political issues have also contributed to this rise, with many community clinics raising co-pay rates.
For gay men, the same CDC report noted that syphilis cases continued to rise, and had been doing so since 2007. Earlier reports from the CDC have noted a decade long decrease in condom use and rise in HIV infections in gay men. The CDC sees the two things as interrelated.
All of this pre-dates the 2012 adoption of PrEP in the US. Only about 20,000 people were taking the medication at the time of the 2015 statement. Furthermore, STI rates have also been rising in Europe, where PrEP use lags behind the US. Whether in the US or other countries, the adoption of PrEP was to deal with the rise in HIV infections — an STI — something that, again, had been occurring for many years.
The question that then follows is: as more people use PrEP to reduce alarming HIV infection rates, will they also use condoms less? One observational study from Kaiser did report 41% of participants said their condom use decreased after beginning PrEP. This was a retrospective study, done after people began PrEP. No self-report of condom use was taken before PrEP began. Studies that did ask and track condom use before and after starting PrEP showed either no change or an increase in condom use. There's a conflict in reports here, which is something that future studies will need to clarify.
In terms of STIs, both the US PrEP Demonstration Project and the PROUD study in the UK found no evidence that STIs increased after starting PrEP, though subjects did have STI infections.
PrEP entails either twice-yearly or quarterly STI checks, depending on the standards a clinic follows. Over the long term, this may encourage people to use condoms more or less, or it may have no effect at all on condom use. Right now, the data just don't tell us that.
Just as the advent of oral contraceptives led to claims that women would become sluts, similar voices are stating that PrEP will result in PrEP users getting more STIs. And while this is a possibility that can’t be discounted outright, it’s not the case right now. Any headline, article, or person that claims it is a fact is simply wrong.