You shouldn’t capitalize the first letter after a semicolon unless it is a proper noun. Treat the semicolon like a comma and follow normal grammatical rules after placing one.

Don’t capitalize words after a semicolon!

Though they have their similarities, a semicolon isn’t a full stop. Just like you don’t need a capital letter after a comma, you don’t need one after a semicolon. Of course, this only applies to ordinary words. If a word always starts with a capital letter, such as the name of a person or a city, it won’t change because of a semicolon.

For example:

• I have been waiting for this day for over a year; Finally, it has arrived. (Incorrect)
• I have been waiting for this day for over a year; finally, it has arrived. (Correct)