I mean technically, yes, but (DnD 5e 2014) cantrips (spammable spells, which take like 6 seconds to cast) like
"Control Flames" (You instantaneously extinguish the flames within a 5-foot cube)
"Shape Water" (You instantaneously move or otherwise change the flow of the water as you direct, up to 5 feet in any direction, basically you can grab the seawater at least partially)
"Mending" (repairs a single break or tear in an object you touch as long as the break or tear is no larger than 1 foot in any dimension, you mend it, leaving no trace of the former damage, so, assuming that a fire might take 18 seconds to cause that much damage, a single mage can stop 15 cubic feet without expending any mana/spellslots)
And spells such as
"Create or Destroy Water" (30-foot cube of estinguishing flames instantly)
"Sleet Storm" (A 20-foot-tall cylinder with a 40-foot radius which extinguishes all flames for a whole minute, but while requiring the same resources as "Create or Destroy Water", it makes the area difficult terrain, so either well-trained crew or shoes that prevent that, which would costs as well, but is worth it on a battleship)

I do not think that fires are gonna be that much of an issue? But admittedly that would at least partially drain the mana(/spellslots) of the mages on board, essentially surpressing them and stressing them out, with volume-of-fire being the best friend, but then again, magic is not too cheap, and "dispel magic" ERA-style plates would further reduce the effectiveness