As the title implies, I need help creating a Variant of a race such as a dragonborn, but I need to know how to do it without it being broken. Can someone please help me?

By "broken" do you mean "doesn't work on dndbeyond" or "too overpowered/underpowered to work alongside other choices"?

I don't know that I can help you, myself, but narrowing that down could help someone else get involved.

The second one

For starters, what kind of variant are you looking to create? Do you already have a draft, or just some ideas?

Do a comparison of the base material and the material you’re planning to put in the variant. Presumably you’re swapping features/proficiencies? Check and see if they feel like the one you want to put in is of similar power/value to the one you’re taking out. 

As Rellott pointed out, variant races don’t add Traits to a base race the way subraces do, they instead offer alternative Traits to the ones the parent race has. Whatever your Variant Dragonborn gets will just replace something that Regular Dragonborn already get instead. In addition, many folks feel that the original Dragonborn are a little underwhelming after 5th-level since their Breath Weapon really doesn’t scale as well as whatever their class features will be. So even if your trait ideas are a  little bit extra compared to OG Dragonborn, it should still not be a problem in the long run.