I think moving fences too significantly in would cheapen the game, and actually hurt it (see next paragraph).
I think one reason strikesouts are rising is because batters are aware scoring is dropping/runs mean more = swinging for a HR then equates to a better proposition than simply trying to manage a 'regular' hit to then get the guy behind you to drive you in. 2 innings/6 outs all spent swinging for the fences is seen as more likely to result in at least 1 run (a HR, or an XB hit that's a near miss HR) than they are worth spent trying to get a run in with a cycle of 'regular' hits. Naturally players swinging away with all their gusto leads to strikeouts when pitchers either know or guess that this is the mindset predominanting in the opposing offense. How many times does a certain game situation lead us to know a certain batter is going to be given nothing to hit & a walk is on offer for him if he'll take it, and yet he swings at garbage to try and be the hero/win the game with 1 big hit. I think it's this latter mindset that's become writ large. Pitchers may be throwing harder, but pitchers - no matter how hard they throw - will always offer up shit before they'll offer up strikes if they know a batter isn't going to accept a walk. Guys swinging for the fences aren't looking to walk. The shift further kills the belief offenses have that they can string together a bunch of 'regular' hits needed to bring in a run, making the big dinger even more attractive to aim for (& hence risking striking out in pursuit of).
As for speeding up the game, there are numerous little things that could be curbed/promoted to manage that. I hardly think many fans would complain about long/er games if increased scoring/increased incidents of games being back-and-forth shootouts were the root cause of such.