Loose dirt can be removed with a Dry Sponge. Grease spots and dirt that has been cemented on are pretty much impossible to remove without damaging the paint.
Use an all purpose cleaner and the two bucket method. Use one bucket for your fresh cleaning solution and the second bucket to wring your dirty water into. You can either use a sponge or the Don Aslett’s Mini Pro with a Wet/Dry Mop Pad.
Use an all-purpose cleaner with a degreaser like our Soilmaster or a Microfiber Cloth. Both should remove the grease spots unless they have penetrated through the paint.
Vacuum the baseboards regularly to remove dust before it gets cemented on. If the baseboards are already grimy, use a heavy duty cleaner like our Soilmaster with a White Backed Scrub Sponge to remove the dirt.
Dust them regularly with an Extension Lambswool Duster. If the dust is thick or caked on, use a water-dampened Microfiber Cloth to wipe each blade, keeping one hand on the top and one on the bottom.
You can either vacuum or use a dust cloth or you may want to do both. The paint won't adhere to the wall if the dust is on them. Before you paint you want to use a coat of PVA drywall sealer.