In the Mario and Luigi series, I fight every enemy I can.
In the Paper Mario games, I tend to avoid the enemies that are not worth fighting because they either have stupidly high defense or attack (Clefts, Chomps, Wizzerds, etc). This doesn't apply to Super Paper Mario, though, because I fight everything there and you actually have a fair chance at avoiding attacks instead of three enemies ganging up on you for 25+ damage a turn.
I used to avoid them before I understood fighting gives experience, which levels up your stats. (Yeah when I was a child I was that dumb.) Now I fight them when I deem the fight's difficulty just right. If my HP is low I generally avoid enemies, just in case they get the first turn and KO me. The Shroobs in BiS OHKO'd Mario & Luigi at full HP because I failed to dodge their first attack, lol. I tend to level up POW in the M&L games the most. But yeah I fight some battles and avoid others. In Sticker Star I avoid everything I can since battles are pointless.
I used to always avoid enemies, especially in Partner's in Time (which I still find a little difficult with some enemies).
Then I found out that I couldn't beat some of the harder enemies because of how low leveled I was. And no I fight every enemy that comes in my way, though I don't fight them if I'm revisiting earlier areas since it's a waste of my time.
When dealing with high defense enemies in Paper Mario, I usually just use an item, Piercing Blow, or Yoshi's Glup and I often study the enemies to see what strategies work and don't work. Only time I run from enemies is during the Doopliss Mario scenario.
For the Mario and Luigi games, I usually just fight every enemy I see to rack up the experience points and coins.