There were 57 recorded deaths aboard Kirk’s Enterprise out of a complement of 430 over the three years of TOS, giving a survival rate of 95.6% per year.
There were 61 recorded deaths aboard Picard’s Enterprise out of a complement of 1,014 over the seven years of TNG, giving a survival rate of 99.1% per year.
The others are more difficult to calculate, as Memory Alpha’s casualty lists appear incomplete or are otherwise difficult to parse. Here’s my best effort:
Voyager began its mission with a complement of 153, “over a dozen” of which perished when the ship was brought to the Delta Quadrant. With the addition of Neelix, Kes, and 23 Maquis from Chakotays’ Val Jean, the total exceeded 150 again by the end of the first season. I’m just going to use 150 as the complement for the entire series.
Filtering through the Memory Alpha casualty list for redundancies and estimations, I estimate about 40 deaths can be reliably gleaned from on-screen events. This gives us a survival rate of 96.2% per year across the seven years of VOY, but with a higher margin of error. If we exclude the deaths from the first episode, the survival rate is 97.6% per year.
Archer’s Enterprise had a complement of 85 upon launch, but amazingly there are no recorded deaths for the first two seasons, meaning we need to include 27 MACO’s in the ship’s complement for a total of 112. There were 13 recorded deaths in the following two years, and one additional death (Commander Tucker) six years later in the final episode, which I’m not including in the total. However, Malcolm Reed claims in the “The Council” (near the end of season 3) that there have been 23 deaths aboard the Enterprise, and claims that this exceeds a 20% casualty rate. 23 deaths out of 112 is indeed a casualty rate of 20.5% Only two deaths occur following this statement for a total of 25. This gives us an overall survival rate of 94.4% per year, but beware the power of averages, as the majority of these deaths are from the Xindi arc. In seasons 1, 2, and 4, during “normal” operation, Archer’s Enterprise had a survival rate of 99.4% per year. During the Xindi arc, the survival rate was only 79.5% per year.
If we ignore Reed’s statement and calculate using only on-screen confirmed deaths, the survival rate is 95.5% per year overall, 99.4% excluding season 3 (unchanged), and 90.2% during season 3. It seems more fair to use these number for comparison this is the only series for which we’re ever told flat out how many people have died and it has a dramatic effect on the results.
None of these figures count non-permanent deaths, cases where the character is permanently replaced by a identical copy, or “deaths” of duplicates/fusions/fissions such as Trip’s mimetic simbiot (“Sim”), Tuvix, or B’Ellana’s Klingon half.
As for DS9… I’m working on it.
If you count the deaths from the Xindi arc, then Archer's Enterprise and Kirk's are practically tied for the most dangerous, while Picard's Enterprise is definitively the safest. If you don't count the deaths from the Xindi arc, then Kirk's Enterprise is the most dangerous, and Archer's narrowly beats Picard's for the safest.