I've been in several dojos across the spectrum, from having a beginners class that focused mostly on ukemi and footwork, to one with a beginners class mostly in name only, to one without a beginners class at all.
In all of them, however, 99.9% are happy to work with beginners, no matter what is being taught. It helps the senior learn by showing, it is a nice break after an aerobic workout with a 20-something ikkyu, it is happily done in remembrance of help given to the giver years ago. the other 0.1% should be smart enough by now to avoid beginners if they really don't want to train with them, so if you grab a senior, s/he wants to train with you.
My first dojo had an open practice once a week, an hour long, when the dojo was open for use without formal instruction. I would go to every open practice to work on ukemi and shikko; once I wasn't a total klutz in ukemi, I would often get a chance to uke for some of the other students who were practicing for their tests (and got a sneak peak that way at some of the more advanced techniques)
I also showed up as early before class as my schedule allowed each day, to work on falls or maybe get tossed around a little. I also found it a good time to ask a senior (one had to be there to open the door) any questions I had on stretching. After class was another option, but not for long as usually folks wanted to lock up and get going.
My first dojo also stressed the importance of rushing to the demo uke for a partner, and it was quite a feeding frenzy to get him; yet one night, as a beginner, I was successful nearly every technique. Now, I might be fast, but I'm not stupid, and I know I succeeded because he chose me out of the melee (having since then sat in that spot, I know now how he managed it)...similarly, other seniors would magically appear next to me at partner time. I will always be grateful to them, and hold those memories fondly, and am happy when I have a chance to do the same for others now in the place where I was...and I think most--from those of us slightly less green to the very old hands---feel the same.