Did you hold him after he drank water subsequent to a dry water bowl? Snakes almost never drip fluids during resp infections unless they just drank water prior to the exam - respiratory secretions during infections are too thick to actually "drip" except in unusual circumstances. Additionally there will be obvious signs of respiratory distress; mouth breathing, head held up often in vertical position, crepitation (sounds of fluids in trachea/lungs in bad cases), restlessness.
Again, if imbibed water (very recently drunk) MIXES with respiratory secretions you may get dripping when the snakes head/neck is held in the down position. Barring this situation there will be mimimal or no dripping. Also, during the restlessness of an active resp infection the snake will smear secretions on the sides/front glass. This mucus will dry leaving residues of mucopolysaccharides/bacteria, and it is infectious. These mps's are what make the secretions thick and in MOST cases too thick to actual drip.
If infections are caught early thermotherapy (raising the temp to the upper limit of the preferred temp range) is usually sufficient to nip the resp infection in the bud. For a BP this would be low-mid 90's. This should be done regardless of antibiotic treatment.
Hope this helps and good luck!
Scott J. Michaels DVM