Agreed with everything DEHF said. Main priority is keeping him in juggles. Even the thing about bthrow and uthrow makes a lot of sense. I have charged a bdacus from dthrow into a good DI'd snake who landed with an air dodge and stopped me cold with utilt before, and it was depressing. I still like to use dthrow untill like 80 % because of the followups it can make. If he doesn't DI at all, you can just run and regrab him. If he DIs high you can start an early juggle. If he DIs low, he will be free from almost any followups, but he will be sent further, and if that is outside the stage, he's in a bigger mess.
Snake wants to recover high and get as close to the middle of the stage as he can. Falco has got great jumps, so he can intercept snake unless he is too close to the blast zone and too far away from the stage, or has got his second jump. When you're used to snake's recovery and falco's jump ability, it will sometimes reward you with the inevitable uair at the peak of falco's second jump after a full hop.
If you miss, snake's got a few different options, depending on how close he is to the middle. If he's close enough, he can space away from you with reversal grenades or reversal c4, which gives him a big change of momentum. You can still juggle him well even with the nades, as long as you make sure to only hit his legs, to avoid hitting the nade.
If Snake recovers and doesn't have his second jump, he will most likely not be able to get that lucky, however. Repelling him with bair works only so well, but gives him the chance to DI high, so I would personally choose bair at low percent to chain him to another bair or maybe a dair, and kill percent to.. well, yeah. Kill him. Dair will always be priority number 1 offstage if you're able, though. If you are good at hitting the cypher, you can spike him anywhere offstage and get away with it, because the cypher gives you knockback with about 180 degrees trajetory angle iirc. That doesn't mean you should be too brave and make yourself look like a fool by dying, though.
It's important to know how far on the stage snake can go at maximum. If you see that he can't go further than the ledge, grab it. I know snakes today still tries footstooling people one the ledge, but if you do a getup roll, that won't happen and snake will only look stupid and fall. If he can reach the first meters of the stage, you can do almost whatever you want. He will try out a few desperate things like nair, b reversal to grab the ledge etc, but fsmash will cover everything.
On stage you will be ok as long as you keep up your game. Snake's dangerous tools on stage can be a pain to deal with. His dthrow can put you in a very awkward position and even lead to an utilt so watch out for his insane pivot grab reach. Be wary of his nades and mines. He can make things difficult for you to approach him, but that's none of your concern. Know the timer of the nade and the hit-, and hurtboxes of your phantasm. While you're invulnerable to the mines at the first part of the phantasm, you can also avoid hitting the nade at the last part, because of lack of hitbox. In short, just space your phantasm with care against snake and not in a rush, no matter how scary he is. C4 and landmines should be harmless if you make a few good mental notes to yourself while watching snake as he put them - landmines are more beneficial to you than to him if you are careful. You can even escape tech chases by rolling through them and make combos by chaingrabbing him into them
His tilts are what makes him so scary, so you have reason to respect them. Always be wary of the utilt, as it is his greatest KO weapon of all. It's disjointed, but should be easy to avoid unless you're in a tech chase. If you're at high percent, be very careful if you decide to get close to him. Camping is definitely your best option here. Have your shield finger ready whenever you are close to him. If you are close enough, jabs would be sufficient. If he ftilts you within jab range, you can avoid the second ftilt simply by pressing the control stick towards him (c-stick would help too, but isn't necessary at this range). If you shield ftilt, you can react a good dash attack for a punish.
This is probably my best matchup, and definitely my most played one. Your jab quite outmatches snake if you're close enough, so this is where you want to be if you're not camping, and you're winning the camping by simply shooting, reflecting, dodging, and an eventual retreating. You have a chaingrab, you can kill snake early if you get him offstage, but onstage snake can rack up damage almost just as quick as you would and he kills much earlier. I'd say the matchup is pretty even, with maybe a slight advantage to falco.