First, for very rule there are exceptions.
Strikeouts assigned to players (pitchers or hitters) are informative as to skill-sets. For a pitcher, they matter most and indicate the type of "stuff" they have. Their "stuff" can indicate whether a batter has a better chance of squaring the bat on the ball.
For a hitter, they indicate more eye-hand coordination and pitch recognition, than plate discipline. Most, not all, high walk types, also have high K rates.
In the context of a game, there is little difference between a strikeout and any other out. Using linear weights, for instance, a runner on 1st base with 0 outs scores about 38% of the time, drops to 25% with 1 out and down to 12% with 2 outs.
Now, assuming an out was "productive", which is rarely the case, the runner moves to 2nd base with 1 out and the chance of scoring rises from 38% with 0 out to 41% with 1 out. You sacrifice an out to gain only 3% better chance at scoring.
Now, given the above, if there is still a runner on 1st base, but there is 1 out. The end result changes. With 1 out, the change of scoring with a runner on 1st base is 25% (see above), but if there is a "productive" out, the chance of the runner on 2nd base with 2 outs drops to 21%.
Now, to actually address all of this, you need to understand the frequency to which they happen. Even a fielder's choice, may not result with a runner on 2nd base. All of these "productive" outs occur less frequently than grounding into double plays.
In 2011 AL
25.7% of all outs were Ks
10.5% of all possible double play scenarios = an actual DP
5.6% of all outs were double plays
2.8% of all outs were caught stealing or pickoffs.
1.1% of all outs were sacrifice flies
1.0% of all plate appearances result in a reached on error
0.9% of all outs were sacrifice hits
Now, to beat some to the punch, obviously it is quite different if we are talking a runner on 3rd base with 0 outs. 5.3% of all Plate Appearances came with a runner on 3rd base with less than 2 outs.
0 outs = 86%
1 out = 68%
2 outs = 29%
Now to be clear, Ks for a hitter with runners on base is a negative. But, that negative is so slight that most sabermatics say "an out is an out". Truly, the base occupancy and out state dictate how negative it is.
Linear Weights (regardless of outs)
MOB 1b 2b 3b hr bb k out
--- 0.29 0.49 0.68 1.00 0.29 -0.20 -0.20
x-- 0.49 0.97 1.36 1.74 0.43 -0.32 -0.36
-x- 0.72 1.00 1.16 1.60 0.23 -0.39 -0.34
--x 0.72 0.86 1.00 1.51 0.21 -0.48 -0.29
xx- 0.93 1.54 1.94 2.38 0.56 -0.52 -0.48
x-x 0.88 0.93 1.77 2.22 0.38 -0.61 -0.46
-xx 1.17 1.46 1.62 2.07 0.23 -0.70 -0.56
xxx 1.38 2.00 2.40 2.86 1.00 -0.82 -0.68
ROB 0.73 1.14 1.49 1.92 0.42 -0.44 -0.42
avg 0.49 0.79 1.06 1.42 0.35 -0.31 -0.30
These values are derived from actual historical data. The conditions were then assigned values based on not just scoring 1 run, but for scoring more runs. As you can see, in any other base state than a runner on 3B, a strike out is not much different than any other out. And again, only 5.3% of Plate Appearances happen with a runner on 3B with less than two outs.
For certain hitters, you accept the strikeouts (i.e. Matt Kemp). That is, if there are other positive skill-sets they bring along with the high Ks.
There have been a few quality/successful pitchers who did not have high K rates (i.e. Kenny Rogers). But, they too had other skill-sets that made it work, usually very good command.
