Lay your 78 RPM record flat on a soft material to protect the surface that is face down. Something like a sheet is best. This will protect your 78 from touching any surface that might scratch it.
Dampen a soft, lint-free cloth with either distilled water or a record-cleaning solution you can purchase in many hobby or record shops. This solution contains no alcohol or acetone, which can severely degrade the materials used to make 78 records, especially the earliest ones. Water is sufficient for cleaning in all but the most extreme cases.
Wipe the record beginning at the outer edge and work your way clockwise, collecting any dirt or dust in the grooves. After each complete rotation, lift the cloth away by turning your wrist slightly inward and lifting the cloth. This helps lift the dirt away from the record. Dampen a new portion of your cloth after every two cleaning rotations to prevent grime from going back onto the record.
Avoid getting the center label damp. The 78 RPM records typically had only a very small hole in the center, for fitting the record on the turntable. The labels on the record are large and easy to wipe against as you clean the inner edge of the record. Even a little moisture can cause the labels (already weak with age) to lift away from the record. For best control, place the cloth on the very tip of one finger and slowly wipe around the outer edge of the label.