5e: Direct information on climate is limited and scattered
Currently the Sword Coast Adventurer's Guide is the best source of 5e information on the sword coast. Unfortunately (for our purpose anyway) its focus is on political groups and mentions of the climate in a given area are rare. The island realm of Evermeet is mentioned as being, "the island
paradise of the elves", and since most of its population is sun elves perhaps we can assume it's pretty nice? We also find out The Cold Lands are , "...in the cold, dry environs of the northeast", which is great but sadly not actually on the sword coast.
There are a couple of maps in the SCAG that we can use to infer about climate. The map of the coast shows areas near the sea of swords to be (mostly) grasslands, a couple of marshes as you move further out, and then the desert of Anauroch even further out. From this we could conclude that a fair bit of rain comes in off the sea and fades out before getting too far inland. The second map is of the north and shows that it is, shockingly, cold.
Alright well how about Hoard of the Dragon Queen. The adventure takes us north along the sword coast so we can look for references to the climate in the areas we pass through. The first mention of bad weather is around the Mere of Dead Men is, "...cold, muddy...", which is great news! The Mere of Dead Men is located north of waterdeed, right on the coast, and hemed in by mountains. Its waterlogged status fits our theory that rain from the sea is a dominant force (being held back by the mountains the rain would be very heavy in this area). Also since up to this point it has been nice enough to mention it being cold we can assume that south of here it's at least mild.
3e: There's a whole book on it
If you're interested in information from other editions then you are in luck! The Forgotten Realms Campaign Setting from 3rd edition has reams of geographic and climate info. A sidebar on page 78 about Waterdeep's climate info discusses its latitude, the nearby oceans moderating aspects on its weather, and the annual arrival of arctic winds. The level of detail available is great and you can easily find information the climate of any specific area you're interested in.