계속 means "continuously, consecutively, unceasingly" 계속 보고 싶다 means "want to watch continuously" or "still want to watch/still miss" depending on the context. ex) 여기에 있는 동안에도 그들이 계속 보고 싶었다. I've missed them while I've been up here. 그는 그 영화를 계속 보고 싶었다. He wanted to watch the movie continuously. (He didn't want to stop watching the movie.)

It can means either "want to see you in the future" or "can't stop missing you." The meaning of the sentence depends on context. Examples) 앞으로도 계속 보고 싶어요. "I want to see you later." When saying good bye after a blind date, someone can say like that. 영화 계속 보고 싶어요. I want to continue to watch the movie. 다른 일을 할 때에도, (당신이) 계속 보고 싶어요. When I do other things, I can't stop missing you.