Yes. People will still have sex. And do. Google "swinger clubs" or "seedy bars".

Attempting to stop people from doing such only drives such behavior more underground and reduces chances for testing and prevention of STD transmission.

(Just as an aside, condoms are not 100% effective)

The biggest issue right now is that there is not enough stigma in regards to being a non-tested person. Everyone even if only occasionally having any type of encounter (condoms or not) should be getting full and correctly done STD screenings every 3 to 6 months. (Where full means multiple diseases checked for and correct means including both throat and anal swabs)

In regards to the statement about "a cure for HIV" I would like to point out that at this time we do have a major weapon against at least HIV type 1 spread called PrEP. There is a pill which if taken once a day, (and you can't skip doses), which is effective > 99.9% at preventing infection. If this were made more widely available, in combination with much more regular testing, while the disease wouldn't be cured or eliminated, the rate of infection could be dropped dramatically. See https://www.out.com/news-opinion/2017/1/13/massive-drop-london-hiv-rates-likely-due-buying-prep-online Note: PrEP is recommended for either sex and for any sexual orientation.

In case you want to know about PrEP: https://www.iwantprepnow.co.uk/