The closest sea, and also the one that bears some relevance to Swiss` commercial interests at an early enough time, is the Mediterranean.
The challenge is tough, but here is my attempt:
A better 15th century for the Swiss might mean instead of a
Zürichkrieg earlier
Ennetbirgische Feldzüge . If this is done at the right time, before Italy becomes contested between strong consolidated centralised kingdoms, the Swiss can play their part e.g. in the Lombard Wars and acquire a little here and a little there. Maybe the
Golden Ambrosian Republic of Milan allies with them and lives a little longer; that would bring the Swiss border right next to the Genoese Republic.
Now, come the alt-Italian Wars, the Swiss won`t just be French mercenaries, they`ll play their own cards, and if they do it right, maybe the Genoese Republic crawls under their umbrella (like they had done with France before) and becomes, through a contract which the Genoese may consider just making a long-term contract on hiring Swiss mercenaries with a few circumstantial addenda, while the Swiss consider it Genoa becoming a Zugewandter Ort.
Then, Switzerland would have inherited all the remaining Genoese colonies and, first of all, Genoese naval power, for if you want maritime expansion, they`ll need to buy into an existing power in the domain, otherwise it´s a bit late to start developing it from scratch.
A Swiss-Genoese naval expansion may begin with the Superba`s traditional focus on Corsica and Sardinia, but it may well begin to develop an Atlantic focus in the later 16th century. From then on, it´s mostly white canvas. They certainly wouldn`t have the population for a settler colonialism, but a few Caribbean islands might work.