I increasingly feel like every matchup is like that for Falco. :/
As far as not getting gimped, I would just focus on these few things:
1. Directional Influence
Make sure you are always consciously DIing and not just going with what you always do. This means being aware of what throws they might use, how far you are from the ledge and plats, and what percent you're at. You almost always want to DI so that you slide off and grab the ledge whenever possible. This is especially important vs. Marth's dthrow. You should even be doing stuff like DIing and techrolling in the first time a Marth dthrows you so that when they inevitably regrab you, you will be spaced to DI the next dthrow onto the ledge.
2. Shine Stalling
First realize that you shouldn't always shine stall. Sometimes you need to choose your action immediately because if you always shine stall, then all you're doing is giving your opponent more time to get into position for when they react to your DJ. Shine stalling works because it messes with their natural timing, which can, for example, cause a Marth to dtilt prematurely, at which point you can DJ and dair right into him (I do this all the time, it's so good lol). Again, just be careful about shine stalling when you didn't necessarily need to because getting hit by a laser or needle and royally screw you over.
3. Double Jump
Your two main options are DJ forward onto the stage/towards the ledge, and DJ back to avoid attacks the opponent may run/jump off with (and needles). Then of course there's a whole bunch of smaller adjustments in between, but I usually find myself sticking to one or the other. Going anywhere in between usually just means you'll be covered by more of their options. The timing of your DJ is also really important. You don't want to jump really early and get your jump sniped by something, but going too low makes it so they can just cover the ledge Phantasm and react to your up-B. One particular thing people seem to overlook is that DJing even slightly higher than the ledge gives you MUCH more flexibility with your Phantasm timing. If the very peak of my jump is just barely high enough to reach the ledge with, then they know exactly when I will Phantasm. If I go slightly above (even so little that Falco's whole body isn't above the ledge), then suddenly the opponent has to cover a bunch of different timings. I could side-B to the ledge as I rise, I could side-B as late as possible as I fall, or I could side-B at any timing in between those two (which can be quite a large window when you are talking about the opponent covering the Phantasm with a Marth dtilt or Sheik needle). Another thing to keep track of for higher side-Bs is what attacks your opponent will use to cover it. Marth's almost always default to jab when you're high and dtilt when you're low. This means if you get the opportunity to DJ just barely above their jab range and side-B, you will probably catch them off guard. I could write a freakin' essay about DJing as you're getting gimped, so I'll just say you need to experiment with a lot of combinations. Overall though, just focus on not getting hit out of your DJ because it's almost always curtains at that point.
4. Phantasm vs. Firebird
Phantasm is obviously better most of the time because of its speed, but Firebird is often underestimated when you consider its flexibility in which direction you want to go. Quite recently, I have found decent success by just up-Bing any time my opponent isn't close enough to react and jump out and hit me during startup. At the same time, however, brute forcing your way back on the stage with Phantasms can be surprisingly effective as long as you keep in mind the tips I talked about above as far as DJing so that you can mix up your timings. SDIing off-stage moves upwards is also crucial to being able to side-B to the ledge where most Falco players would be forced to up-B. Mango does this constantly. Knowing when you can get a hit off with a recovery move is also really important to note as you go off stage. Much like with DJing, there's just a lot of scenarios that are dependent on the opponent's character, your percent, and all of the steps before this one.
5. Don't Get Regimped
Making it back onto the ledge or stage from a gimp attempt is all well and good, but it's pointless if you just get thrown/hit right back off into a similar situation. Always make sure you're minimizing your damage as much as possible by ledge cancelling Phantasms, Firebirding to the ledge instead of the stage when possible, ledgedashing with invincibility as soon as you touch it, and not doing predictable rolls or spot dodges if you end up holding shield by the ledge. Worst case scenario, make sure you are ground teching or SDIing their attacks to knock you back off so that you have more time and distance limitations to work with for your next recovery attempt. Sometimes, it's okay to up-B/side-B onto the stage if you know that once they hit you back off you'll be able to Phantasm to the ledge before they get over there. There's a lot of similar things like that.
6. Figure Out What Went Wrong
Since so much of learning in Melee is about trial and error, being able to properly determine when and how something works is very important. First, realize that recovering is inevitably not always going to be in your control. Sometimes, even with a bunch of good, subtle fakeouts and the perfect spacing of your actions, you may still get gimped and die. This is just the nature of Falco's awful recovery, and Melee in general. After all, if we could master recoveries to the point that we always got back, we wouldn't be so worried about getting grabbed by the ledge in the first place. My point is that just because your course of action didn't work doesn't mean it wasn't the best decision. When you are relying on the opponent to mess up a timing or spacing in order for you to get back, you may only get back 3/10 times with the same exact technique. If every other approach yields less than 3 successes per 10 attempts, however, you obviously shouldn't stop doing it. I'm also not suggesting finding one thing that works and sticking with that though. Being unpredictable is Falco's greatest chance of recovering, so instead of thinking about whether you did your recovery right, also consider that perhaps your opponent just expected it. Figure out why they are expecting certain recovery tactics and which ones they aren't expecting, then execute those. Maybe they can tell you will DJ back if you DIed away or DJ forward if you DIed in. From there you'd have to figure out how to alter it. Maybe all you need to add is a shine stall, but it also might not hurt to occasionally DI in and still DJ backwards. This cycle will never really end because you'll always have to adjust your decisions to your opponent and what recent decisions you've made that they've adapted to.
