I do not see a difference between placing an order for stock or (crypto-)currency. Maybe the question arises because crypto-currency trades 24/24 7/7 unlike certain stocks.
The question of putting in a limit order which might or might not get triggered on Shabbat was asked before and answers tend to be lenient, i.e., it is allowed if (1) there is no human intervention post-trade and (2) you do not explicitly plan for the transaction to happen on Shabbat (as in a time-delayed order). Both conditions are satisfied with crypto-currencies.
But as always, CYLOR before acting on anything you read here.
dinonline writes (regarding Yom Tov since, unlike Shabbat, stock trades on Yom Tov) - the case here is after the fact, i.e., a limit order was triggered after the questioner forgot it
Although it would not be correct to intentionally make an order for
purchasing stock on Shabbos or Yom Tov, post factum there is no
problem in keeping the stock, and this would not be considered an
issur, certainly if done without intent.
Sources: Authorities dispute whether it is permitted to arrange for a
kinyan to take place on Shabbos. Rabbi Akiva Eiger (159) cites a proof
from Terumas Hadeshen (269) that there is no prohibition is doing so,
but continues to prove that it is prohibited. Iggros Moshe (Orach
Chaim III, no. 44) defers both proofs, and writes that one should be
stringent, and this is also the opinion of Radvaz 456. However, a
number of authorities are lenient in this matter -- see Shaul Umeishiv
VI, no. 50, and Divrei Shaul, Shabbos Hagadol, concerning selling
chametz on Erev Pesach that falls on Shabbos; Maharshag 13; Maharam
Schick 131; Machaneh Chaim, Orach Chaim III, no. 22), and see also
Avnei Nezer (Orach Chaim 51).
Thus although in deference to stringent authorities, it is not proper
to intentionally arrange a sale for Shabbos, as a rabbinic prohibition
(making sales on Shabbos is a rabbinic prohibition) in doubt, it
cannot be called a transgression. Certainly, if done without
intention, no issur can be said to have been transgressed (according
to Nesivos Hamishpat, even a certain rabbinic prohibition transgressed
without intent is not considered to be a transgression), and no
repentance is required.
Eretz Hemda writes similarly
Usually, a non-Jew who is paid by the job may do such work for a Jew
on Shabbat, as he is considered to be working on his own behalf for
the pay. However, he may not be told to do the work on Shabbat
(Shulchan Aruch, Orach Chayim 247:1). In this case, your instructions
are that if the limit is reached on Shabbat, they should make the
transaction specifically then.
We have ascertained that after you place the order by computer, there
is rarely human intervention in the processing of your trade. When no
melacha is being done on Shabbat on your behalf by a person, but only
automatic computer activity, the main question disappears. Someone at
your brokerage may do melacha in sending you confirmation
notification. On the other hand, even if this does happen, you do not
need to ask them to do so on Shabbat.
See there for further sources on time differences between the location of the client and the location of the stock exchanges - not directly relevant to your question.
Finally R Yosef Y Kushner in his book Commerce and Shabbos writes regarding buying/selling stocks on Shabbat/Yom Tov via automated programs
One should avoid specifically scheduling the sale to take place on Yom
Tov. [...] It would be permitted to program a sell or buy order based
on a particular price and not dependent on any specific date. [...]
This is because since the sale was not scheduled specifically for Yom
Tov there would be no issue of doing business on Shabbat (issur
meka'ch u'memkar). However one must be certain than no human
intervention whatsoever is required to execute the sale thereby
bypassing any issues related to amirah la'akum (telling a non-Jew to
do work on Shabbat).
Update (May 2021): R Daniel Mann (here) has now dealt with a very similar question focused on crypto-currencies as well, and tends towards leniency. See there at length for his reasoning and sources.
