Defining a ship
If defining a ship as having a minimum of one human crewman...
In the absence of FTL drives, a minimum for an active human crew is however many are required to perform the maintenance tasks on the life support, drives, and power systems.
Since this is extremely variable based upon automation and durability, what can be said is merely "you need engineering crew."
Human need to interact
Given human need to interact, this sets a further minimum crew standard. The military has had a number of small outposts for years. Minimum crew size for these correlates directly to maximum duration of assignment. A 4-5 person crew tends to start experiencing issues between 2 and 6 months deployed in isolation, even with routine work contact via radio with outside persons. A 20-30 person crew can go for 4-6 months before similar issues. After this initial period of troubles, many crews are able to work things out, and a stable long term status quo evolves; but when it doesn't, the results are often tragic. NASA long term missions have typically been 3-5 person crews, and the crews have been put in isolation for the requisite several months during training; those who can't handle it have been weeded before flight.
In looking at small communities, the smallest psychologically stable communities appear to be in the range of 20-30 people; smaller communities (such as McGrath, Alaska) tend to be stable until some stress, then violence happens; usually a new stable evolves shortly thereafter. The problem being that a spacecraft can not afford such violence.
So, for an STL ship, you'll want at least 20 active persons, rotating which 20 if practical, in order to maintain the hamlet level of community which seems the minimum for long term psychological stability.
Colonial Genetic Stability
Since any mission to another star system in an STL craft is likely to be one-way, the overall mission crewing needs to be considered as well. And the limit there is the genetic viability of the colony. If completely unrelated individuals and absolute breeding authority can be presumed, as low as 30 individuals may be viable. Long term, the lowest genetically stable populations in isolation seem to be about 100 individuals - a small village level, or several hamlets. At about 200, consanguinity selection for the colonists sent is a low priority.
FTL - Alcubierre-White Warp Drive
Assuming Dr. White's assumption of 10C capability is correct, this puts many nearby stars under a year away. Within a year, the technical needs of the hardware trump the psychology of the crew. Keep them busy, and each with private space, and a crew of 3-5 should be functional long term. Even up to 2 years, to allow for a round trip, a Warp Drive at 10C can reach any of a dozen stars, specifically, any of 9 systems.
We can't yet estimate the requisite engineering/maintenance times.
We can note that, unlike STL ships, not all are one-way trips. Therefore, the colonial need is also able to be ignored.
FTL - other
There is no other FTL drive thought feasible at this time. Therefore, no practical considerations can be made.