The direct answer is that it happens in the broader prison system, but it is a serious federal crime and the consequences for the staff member are severe when it does.
Under the Prison Rape Elimination Act, any sexual contact between a staff member and an inmate is classified as sexual abuse regardless of whether the inmate appears to consent. Consent is legally irrelevant in a custodial setting because the power imbalance between a corrections officer and an inmate makes genuine consent impossible under the law. A staff member who engages in sexual contact with an inmate is not having an affair. They are committing a felony that typically carries five to eight years in federal prison, loss of their career, and placement on the sex offender registry. The inmate becomes the victim and the officer becomes the defendant.
Facilities take PREA violations seriously because the legal and institutional exposure is significant. Reporting mechanisms exist specifically for this reason, and investigations do result in terminations and prosecutions.
Now for the honest part of this answer. If the question behind this question is whether your inmate is involved with someone on the staff, that fear lives in the same place as every other fear that incarceration puts into the heads of people on the outside. The isolation, the lack of control, the imagination filling in gaps. It is real and it is painful, but acting on that fear by cutting off contact punishes you more than anyone else.
If the relationship has reached a point where this is where your mind is going, that is the conversation worth having directly with your inmate rather than through a question about facility policy.