As far as I know, every big city has psychoanalytic institutes, sometimes more than one. They might have their own counseling centers where you can get psychoanalysis at a low cost or even free of charge if you commit to a certain time frame and a certain frequency of sessions. In this case, you'd be seeing a psychoanalyst in training, but they are a licensed professional, because in the US you can't start the psychoanalytic training if you are not licensed yet. Psychoanalytic training goes on top of the standard academic program (for those who want to pursue it) and it's always post-licensure.

In my city, all psychoanalytic institutes have clinical research as part of their training program. Each analyst in training is supposed to have several clinical cases to conduct a research. If you agree to be a research subject, which would entail seeing an analyst 3-5 times a week for one year minimum, you could get your analysis for free.

Other than that the only reasonable way to afford psychoanalysis is to find an analyst who happens to be a provider from your insurance, in which case you'd only pay a co-pay. But they are rarely on insurance panels. Their classic way to build up practices is to form and maintain consistent sources of out-of-pocket referrals. They maintain strong connections with psychiatrists and other MDs, family lawyers and other professionals who could refer clients who are able to afford their fees. Many analysts are also pdocs and are well connected to other medical professionals. That's how they get a well paying clientele.

All in all, psychoanalysis these days is a model that is largely for those who are well off, not a middle income and certainly not a low income folks.