I am still learning and experimenting with different ways for every thing. The only time I have had issues with running is when I used my Ink Jet to over print Wine Expert pre-printed labels. They seemed very prone to smudging. After letting them dry with out rubbing for several days they were fine, and test washings when removing labels show they stay good. My latest batches have been following best forum advice all the way. Using Laser Printing at Kinkos, then spraying with Krylon 0500. Those labels I can scrub with Scotch Brite Scrubbies, and they stay Pristine as the day I thought them up.
Specifically, to your question. Ink Jet on plain paper has been fine after drying. I haven't seen any issues with ink running under water. I do have problems with plain paper uncoated with Krylon 0500 falling apart under water. I use some ink jet on plain paper. My biggest issue is print quality resolution. The intricate subtle detail I spend hours creating, ( that no body notices anyway), is wasted on ink jet, but shows beautifully on Kinkos Laser.