Can I use Mint.com without giving out my bank details (credentials), and manually enter my transactions instead?
You can, but there's really no point. You might as well use a separate money tracking program.
The main advantage of Mint is that it aggregates financial information from multiple sites. For instance, if you have a car or student loan with one bank, your checking account at another bank, a 401k with your employer, and a credit card from a local credit union, Mint can pull all of those accounts and display the summary on a single page (or allow you to view details on each account individually).
If you're manually entering transactions, there are superior programs available.
I use mint.com and notice that you can add transactions manually. That is how all the information is aggregated and is also how trends are created.
So based on that, you could manually categorize everything and would be able use it to manage your finances. The one thing that I'm not sure of is how you would add a savings/checking/investment account. To add those types of accounts, you need a url of a bank that you can enter information for.
Note: If Mint.com requires that you enter a bank account, you could enter a dummy bank account that just wouldn't connect - it is a sloppy work around, but it would stop any request to create an account (You might not even need to do it).
Let me know how it works for you if you decide to use mint without connecting to any sites. I hope this was helpful.
EDIT 8/9/11: Mint has just added the feature to iPhone app to keep track of money spent. You can use this feature to just manually enter your transactions without giving them your bank details.
As others have pointed out, you can, but there's really no point in doing so given the other options available. Specifically, I'd recommend just using Google Sheets if you don't want to give out your banking information as it's easy to use, access, and develop great reports with. Additionally, you could upgrade over time to something like TillerHQ which is a great combination of the two.
Note: I was a former Mint user that gave up because their sync technology was constantly breaking. Moved to TillerHQ and it's been great.