This patent expired in 2000.
Rivest, Shamir, and Adleman were assistant professors at MIT, paid primarily to publish papers. They freely discussed the RSA cryptosystem with Martin Gardner for publication in his Scientific American column in August 1977.
Garfinkel's PGP book says (page 78) that the "head of the MIT Laboratory for Computer Science reviewed the RSA research and decided that the algorithm might be patentable. Under the terms of the U.S. research contract that funded the lab's work, MIT was obligated to file for a patent on patentable research. So that fall, Rivest worked with the MIT patent office to prepare a patent application for the RSA algorithm ... The patent gave MIT the right to stop anyone within the United States from making, using, selling, or importing devices that used the RSA algorithm. However, since the algorithm had been published before the patent application was filed, MIT could not secure foreign rights to the invention."
There's no evidence that Rivest, Shamir, and Adleman were thinking about patent monopolies when they introduced the cryptosystem. On the contrary, the evidence shows that they freely distributed the idea (as one would expect from academics), and only later were told by a manager to patent the idea.
Meanwhile there were people other than Rivest, Shamir, and Adleman who had seen the DH public-key cryptosystem and were searching for further examples of public-key cryptosystems. One of these people, Michael Rabin (1976 Turing Award winner, before that rector at Hebrew University), mentioned in a 1978 paper "Digitalized signatures" (on page 156) that a public-key system "employing large prime numbers was discovered by the author (unpublished) and independently by Rivest, Adleman and Shamir [3]".
This is an example of a phenomenon highlighted by Lemley in "The myth of the sole inventor": "Surveys of hundreds of significant new technologies show that almost all of them are invented simultaneously or nearly simultaneously by two or more teams working independently of each other." In the RSA case, with documented discoveries by two academic teams, it is hard to imagine how standard academic incentives would have been insufficent to trigger publication.
Patent 4405829 was actively enforced. For example, Garfinkel's PGP book says (page 100) that "Bidzos' most effective weapon against the organized distribution of PGP was the RSA patent. Whenever Bidzos learned of an organization that was distributing copies of PGP, he wrote it letters demanding that it stop. CompuServe and America Online were both forced to take copies of PGP off their systems. Bidzos also went after universities, demanding that they not make PGP available to their students. According to Rotenberg, even the esteemed EFF took PGP off its FTP site."
A technical report from Rabin in 1979 described what's now called the Rabin system, which is similar to RSA but has some advantages. An open-source implementation of the Rabin system by Mark Riordan in the early 1990s was shut down by legal threats, even though it isn't clear, as a legal matter, that the Rabin system is covered by the RSA patent. In general, the RSA company's strategy appears to have been to stop most usage of RSA and all usage of other public-key cryptosystems, while promoting usage of RSA that would provide income to the company.