How do I see active SQL Server connections?

You can use the sp_who stored procedure.

Provides information about current users, sessions, and processes in an instance of the Microsoft SQL Server Database Engine. The information can be filtered to return only those processes that are not idle, that belong to a specific user, or that belong to a specific session.

SELECT 
    DB_NAME(dbid) as DBName, 
    COUNT(dbid) as NumberOfConnections,
    loginame as LoginName
FROM
    sys.sysprocesses
WHERE 
    dbid > 0
GROUP BY 
    dbid, loginame
;

See also the Microsoft documentation for sys.sysprocesses.

Apart from sp_who, you can also use the "undocumented" sp_who2 system stored procedure which gives you more detailed information. See Difference between sp_who and sp_who2.