Am having a case of OCD, so I went around Gensokyo collecting any puppet I did not get before the pre-Makai run. What I have found is this:
1) Using the strongest (and purchasable of course) seal, Contract Thread has a great chance of sealing most puppets up to level 40-50 without even having to put its HP in the red area. This means usually one use of the thread can instantly get you the target even at full health.
Past 51 however, even the said Contract Thread does face difficulties. Before I finished the game, I tried getting a level 56 Shinki at the Makai town area. It took me nearly 10 threads even when Shinki's health was on the red line (took me
. Then later during post-game, I tried getting a Speed Suwako at the Giant Toad pond and that alone took me 10.
2) Certain puppets can help a lot when you are puppet-hunting. One of my main puppet hunters (post-game) was Defense Momiji. Using her ability Indomitable , which prevents 1-hit kills, I was able to grab Kasen and Yukari via said ability and her attack Balancing (wiki name, turns both puppet's HP equal regardless of typing except Nether and Illusion).
3) The safest way possible to grab a puppet is via knocking its HP to 1. This is a bit hard, however (with the exception of any puppet that has OHKO-guard skills such as Indomitable and Fair and Square) so the best is that you knock the opponent's HP to the red line.