If the user will provide their login details (username and password) then you just need to provide the ability to enter them in your app, e.g. show a dialog asking for these details. You can then use those values the user gives to build the connection string in your code.
Alternatively, if all your users are going to be using a single SQL account to connect then you can put the connection string in your app.config file using encryption if you want to hide it from your users, see cmsjr's answer for an example of how to do this.
Alternatively, if you're developing this on an internal domain (intranet) then switch your database to integrated security and put your users domain accounts into the relevant access group on your database server. Then you won't have to worry about collecting username or passwords at all.