Our first game was a blast. The only thing - it was too simple for 5 investigators. First scenario was too "introductionary", a piece of cake. And here I was, thinking that basically grabbing more threat tokens will fix and balance the issue. It just didn't happen. So, now I want to ask the community - what is the best number of players for each of the scenario. This applies for Print-on-Demand Expansions and Forbidden Alchemy too. Hope I can get answers before our next play.

It says "to 4" players on the box. Best number in my experience, regardless of story, is 3-4.
3 in stories which rely on mythos cards, 4 in stories with many monsters.

If you play w/ more than 4 investigators, others suggested drawing 2+ threat tokens per additional character. (5 players = 6 tokens, 6 players = 8 tokens, etc.)
Additionally, you will need to fix some rules. Only 1 Maniac per game, with a board full of characters is way too easy.

I've squeezed extra people into two maps before: one was The Inner Sanctum (5) and Blood-Ties (5 or 6), and I Keeped both times. I won. Very easily, and this was with only 1 threat per investigator, so I don't believe there needs to be a threat adjustment from what I have seen.

Your question is what is best amount of players for different scenarios, but to whose favor, or just for balance's sake?

So far, I cannot answer that question because I haven't lost as Keeper yet and when I do play I beat or tie game about half of the time. Naturally the game is just hard for the Investigators, and I haven't seen any rhyme or reason to make it easier for them because the "1 threat per" allotment is quite well balanced I have found in maintaining the game's extreme difficulty for them. Most of the time, it's either "close but no cigar" or they won by the smallest window of opportunity (usually all thanks to good dice rolling and well-devised strategy).

Question: do you play with the variant in the back of the book that lets the Keeper gain a skill token for each investigator playing? We do, and, well, you see the results (honestly I don't think it matters much though; I don't even use my own skill points as Keeper unless I really want a monster to special).