Briefly, crypto mining refers to the use of complicated mathematical algorithms by computers to eventually create, or mine, a fraction of a particular crypto-currency. Typically, when crypto is being mined, the computers stay on for an extended period of time, which likely includes Shabbat.
Halachically, earning money on Shabbat, also known as S’char Shabbat, is problematic (O.H 306) even for acts which are permitted on Shabbat. If, however, money is earned over a period which includes Shabbat, but other days as well, there is room to be lenient. Indeed, the Magen Avraham (ibid 7) writes that if one earns interest in a bank only on Shabbat, then it is prohibited. However, if one earns interest on Saturday (or Friday for that matter) which includes some Shabbat and some Yom Hol, then it would be permitted.
Rabbi Haim Benattar (Hefetz Hashem, pg.18a) writes that the prohibition of S’char Shabbat only applies when work, even permitted work, is actively performed on Shabbat itself, and Rabbi Haim Soleveitchik (Hidushei Hagrach Shabbat 19a) concurs. It would appear, therefore, that currency that was mined on Shabbat is permitted since it is a passive type of process and not something that one is actively engaged in. Furthermore, Rabbi David Ortenberg (Tehila LeDavid, Hilchot Shabbat, 314) writes that S’char Shabbat only applies to earned money and not money that was traded. In crypto mining, the miner receives the fractional currency in exchange for the algorithm calculated by the computer and thus it does not have the status of being “earned”.
Summary: One may allow one’s computer(s) to mine cryptocurrency on Shabbat.