What OTC medications or remedies are safe to use for a runny nose during pregnancy?

Only a few. For allergic rhinitis Nasalcrom or other products containig Cromolyn Sodium (sodium chromoglycate) are same but you have to use then 4 times a day. Nasal sinus rinse will help remove irritants, allergens or obstructing mucus nasal decongestant sprays can be used if totally obstructed but you run the risk of worse rebound congestion.Do not use more than 3 days.
Antihistamines. Otc antihistamines such as Benadryl (diphenhydramine) as well as longer acting nondrowsy formulations such as Claritin have been standard recommendations for years. However if you want to avoid medicines cleaning out the nasal passages with a neti pot or saline spray is also helpful and flushes out allergens such as pollen as well.