It's actually not too far to walk.
Otherwise, service 381 (towards Peckham) runs along York Road (which is off to the left of Waterloo Station, as the train comes in) and then passes along Southwark Street (which is as close as you can get by bus to Tate Modern).
However, my personal preference would be to approach the Tate Modern from the opposite side of the river. On arrival at Waterloo Station, turn right as you enter the concourse from the platform. You'll see the stairs to Waterloo East station on your left. Just slightly further on, you'll come to an escalator taking you down to Waterloo Road. You come out onto Waterloo Road alongside the bus stop for service 26 (towards Hackney Wick). Alight at St Paul's Cathedral. Cross the road and head for the river (which is off to your right as you arrive on the bus). This brings you to the Millennium Bridge. Tate Modern is on the far side.
Chris