I own a small business and we've gotten to the point where I need to hire a couple of contractors to do some work on my LI profile; my website, etc. but I don't want to just carte blanche give them my personal passwords. How have most folks handled that or is it that I just need to get comfortable with hiring a solid contractor to do this and just trust?
There are many ways you can handle this.
Regarding LinkedIn specifically:
You can refer to their documentation about the proper way to work with designated assistants who are given limited access to do the things you need them to do.
Speaking generally, not about LinkedIn:
I am a high-level freelancer with an extensive track record. To be completely rank, clients typically provide me with acces to EVERYTHING soon after they meet me. But I often work with more "junior" developers and team members, and it is common to set up varying levels of access.
One common and very safe way to handle this is to create a development server (or multiple development servers) which are NOT the same as the "prime" server. You can do this for both your file/web server AND your database.
This is a good practice anyway, just for the sake of being able to work on a development system without directly impacting the "live" system.
You can create a development web/file server AND a development database server. And you can populate key data tables in the development server ONLY with "dummy data" / simulated data.
Your development team can create custom functions that create as many simulated data records as you want to have in the database. And none of it is "real" data, so it competely safe to provide to developers.