There are many people here who come from an Electrical/Electronics or related dicipline, including all three of the Admins, so for a start you should not feel that this is anything of a disadvantage in getting where you want to go. Any knowledge gained in either area will be useful, maybe not directly, but will stand to you in the long term.

As for the specifics of this case, cbkihong has already indicated that most modern websites are in fact web "applications" sitting in front of databases of varying size, from the small site where someone has their own blog and nothing else, to huge corporate sites to online shops. In order to be an effective web designer you will need to know how to interact efficiently with databases in order to develop effective websites, so my advice would be to take the database course.