sql server determine total number of open/active connections

This shows the number of connections per each DB:

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

show total Connections

SELECT 
    COUNT(dbid) as TotalConnections
FROM
    sys.sysprocesses
WHERE 
    dbid > 0
    

If you need more detail, run:

sp_who2 'Active'

Note: The SQL Server account used needs the 'sysadmin' role (otherwise it will just show a single row and a count of 1 as the result)