I will assume a "grab and try and pull you down" is some sort of clinch or modified clinch. Sure you can side step. O Soto Gari comes to mind. (Am I making the wrong assumption?)
I had a challenge match against an Aikido black belt. When I applied the Gracie clinch, he tried to side step and do O-soto-gari. What he did not know is that the Gracie Jiu-jitsu clinch done properly has the Jiu-jitsu practitioner on the side of the adversary, hips lower than the adversary, and both legs are squeezing one of the legs of the adversary. So, if I am clinching my adversary and I am on his left side, then I am trapping his right arm in an overhook and trapping his left leg by pinching it with my knees. The leg trap from the clinch keeps the adversary from doing Judo throws like Uchi-mata (which most Judo guys try first when I lock on the clinch), Harai-goshi, and of course O-soto-gari.
Like I said, when I locked the clinch on he then, the Aikido black belt, tried to counter with O-soto, but it was too late. The throw attempt only caused him to stumble, and as he tried to keep from going down I threw him with Tani-otoshi, took the mount, then pinned him until he submitted (you don't have to punch a guy into submission if they do not know the escapes--you can simply apply one of several modified / suffocating pins--much more humane).
If you are going to use Aikido to stop Jiu-jitsu, then you must know what Jiu-jitsu has in mind for you. Jiu-jitsu has a plan, and they usually follow it to the letter. We call this plan text-book Jiu-jitsu. You can go to my web-site, www.kingsportjudo.com
, and read a detailed analysis of the Jiu-jitsu fight plan under the 'Newsletter' button.
You need to study Jiu-jitsu tactics and techniques in order to get an idea what they need to win. Only when you have a clear understanding of your adversary's abilities can you begin to formulate counter-measures.
I teach Aikido, Jiu-jitsu, and Olympic Judo. I can tell you from experience that what makes Jiu-jitsu so effective is that the Gracies spent time learning how other Martial arts work, what they need to work, and then developed simple tactics to spoil their techniques. The Gracie clinch, for example, is an amazing tool. If you can acquire it in a real fight, then it shuts down boxers, kickers, Judo throws, free-style throws, and Greco-Roman throws. Once the clinch is established, you have to carefully pummel, that is a wrestling term, not to be taken in its boxing sense, your way to a more advantageous tie-up. Aikido students should learn basic wresting skills: how to pummel, how to shoot a double, how to shoot a single, how to sprawl when the guy shoots, how to use a front head lock, ect. Aikido students should also seek to learn some basic Judo. Knowing how to throw with the jacket, and how to block Judo throws is helpful. Most Judo throws resemble Aikido projections, so it will only help you.
I usually ask that my students take Judo and Jiu-jitsu for one year, before starting their study of Aikido. I have found, and my students have found as well, that Aikido is very clear to them after having a background in Judo, Jiu-jitsu, and ever Greco-Roman. I love Greco-Roman.
I do agree that proper posture and movement and foot work can prevent this from happening, however it still does happen and we need good solutions to recover from it.
Again, if you are good at pummeling, then you can keep Jiu-jitsu guys from taking you down. You can't keep them from jumping guard. If they can get their hands on you, then usually they will jump to closed guard, butterfly guard, or even half-guard. So, you need to also learn how to open the guard and stand up and back away. Marc Laimon did this successfully against Ryron Gracie when they fought last year. Ryron could not keep Marc from getting up and backing away, which caused the fight to go back to the standing position. But, keep in mind that Marc Laimon is awesome at Jiu-jitsu. Cobra-Kai Jiu-jitsu rocks!
Again, if you are going to beat Jiu-jitsu, then your going to have to study it thoroughly.
If you are doing Aikido for a hobby, then maybe you don't need to be interested in learning Jiu-jitsu or Judo. But if your interested in practical self-defense, then why not study Judo and Jiu-jitsu? There is no law in our land that prevents you from training at more than one dojo. Most guys I know that teach Judo also have a dan rank in Aikido as well. The three, Judo, Jiu-jitsu, and Aikido go nicely together. Have fun!