Of course this will vary from place to place with every town and city having their own specific rules. In general you should assume that at least some of the data you file on your application will be a matter of public record.
For a specific answer simply ask the clerk with whom you will be filing the application.
Just curious, why do you need a business license for a home based internet services company? Is that a specific requirement where you are located?
When you start a business, you can either run it as a sole proprietor or as an entity like an LLC or Corporation. The LLC/Corporation is created by filing organizational documents with the state government, typically the Secretary of State. Those documents are nearly always a matter of public record.
On top of that, the municipality where you live may require you to have a business license. The public records law of most states will require that license to be a public record as well, but JonnyBoats is right -- check with the municipality to be sure.
Even if the license is a public record, though, it may not be easy to find. Some public records are available on the Internet, and others have to be specifically requested. Clearly, records that have to be specifically requested are going to give you more privacy.
If it's a big concern for you, getting a business address really isn't that big of a deal -- a USPS mailing address may suffice, or you can "rent" a virtual office. You still need to check the requirements of your filings, though, to be sure that you're giving them the correct information.