Who decides how many candidates and which of them will be nominated by the Democratic party in a presidential election?
In the United States, candidates are self-nominating. So as many candidates as want to do so can appear. There is no upper limit. This is not specific to the Democrats but true of the Republicans as well.
There are some limiting factors. To appear on the ballot, a candidate generally has to gather signatures from registered voters. That is done separately in each state and candidates don't have to appear in all states. However, I don't believe that there are any states that have started that process yet. So no one has succeeded in getting on a ballot. Nor failed. Right now, candidates self-nominate.
There are some informal limits. For example, some people will be invited to debates and some won't. That will be determined by those who manage the debates, traditionally the national parties in the primaries. So you could assert that the national parties choose who participate in the debates and thus who is considered a serious candidate. The news organizations also tend to ignore certain kinds of candidates as gadflies.
According to Politico:
For its first two debates this year, the DNC said a candidate may qualify for the stage either by reaching 1 percent support in three separate polls — including national polls or early nominating state polls — or by meeting a grass-roots fundraising threshold.
The 2016 general election threshold was 15%. But as a practical matter that only counts if Andrew Yang runs as an independent in the general election, not as a Democrat in the primary. Regardless, this is a threshold for appearing in debates, not a ballot threshold.
Only one candidate will be the Democratic nominee for president in the general election. That person is officially chosen by the convention but has traditionally had some claim to be the plurality winner of the popular vote in the primaries. In 2008, two candidates, Hillary Clinton and Barack Obama, could claim to have won in the primaries. This is because a state had limited votes due to moving their primary dates earlier than allowed. So Hillary Clinton won the vote including that state (Michigan) and Barack Obama won without it.
My point is that if you are asking how many candidates can seek the Democratic nomination officially, that's unlimited. Only one candidate will be nominated by the Democratic party. That person will almost certainly have a claim to having won the popular vote, although it's technically the convention that chooses.