No worries about the rain! Just make sure you dress appropriately. Seward is a temperate rainforest, so it does rain often, and we rarely cancel due to rain. However, it's almost never a downpour. Typically the rain is an on-and-off drizzle that doesn't affect our hikers when the right gear is worn.
What's the right gear? Rain shells! Goretex is amazing stuff, it's super packable, and it will keep you dry in the wettest of conditions. You're coming to Alaska — invest in a good set of upper and lower rain shells and you will be super happy that you did!