Rough-up the old varnish with a #000 steel wool, being careful to work the steel wool into any curves or crevices. Do not work the steel wool too vigorously; it's not necessary remove the varnish--you simply need to remove its shine.
Sponge down the entire surface with TSP or a TSP substitute to thoroughly clean off the steel wool residue and open up the pores of the varnish. Wipe the TSP off using a clean, wet sponge. Make sure the surface is completely dry and dust-free before moving on to the next step.
Apply a water-based bonding primer using a synthetic bristle brush. Look for a primer that highlights its bonding quality in the product's name or promotes bonding as a primary feature on its label. Allow two hours for drying. If the primer appears uneven after it has dried, apply a second coat.
Apply two coats of acrylic latex paint using a high quality synthetic fiber brush or a foam mini-roller. An acrylic latex paint will adhere better than a conventional latex and form a hard, water-resistant finish that should last for many years.