Palms are tropical plants. Different cultivars have different tolerance levels for cold. Is that tree in a pot? If your winters get below 50F, you'll have to bring it indoors for winter. Cutting the bottom growth will not affect the top growth.

I think this is a cordyline rather than a palm, in which case you can cut off the bottom growth if you don't want it. Wait for this to be confirmed though as I've never grown one myself.

I also think that it is cordyline. It is possible to pull the growth from the base and pot on for new plants but your basal growth is quite extensive. Removing from base won't necessarily help the top. It is likely that it sent out the base growth in that really bad winter of 2010. Many cordylines appeared to be lost but if they were left many started to shoot from the base. I think you can only try removing the base growth and see what happens. Removing a cordyline with a stem that thick will take some doing as it is fibrous and clogs normal saws.

I remember your asking about this tree back in July, and we had an extensive 'conversation' about it in that thread.

You'll need a good, clean, sharp pruning saw, or a good pair of loppers (depending on whether the growth at the base has formed woody stems or not) and you need to wait till April or May. Then go out and saw off the extra growth near the bottom as close to the main trunk as you can get, leaving a little collar of quarter of an inch at most, no more. Growth which is just green leaves from the trunk can be sawn through in the same way.

Provided the top of the tree isn't damaged, this might/should encourage new growth at the top. If it is damaged in any way, allow the largest of the basal growths to continue to grow, sawing down the main trunk to ground level, so it can take over and grow into a tree.

Thank you very much for your answers, I'm tempted to just leave it and see what happens but I'll probably end up cutting the base growth around March time.