How to deal with a colleague who always puts you down?

I favor the Confusingly Positive/Neutral Response in these situations.

• "So-and-so did Awesome Thing and you --" "Wow, that's great! When will it be published? I'd like to read it."
• "What was your undergrad GPA?" "Uh, I don't even remember. It got me in here, though, and I love it here!"
• "Bah, what you're doing is garbage." "I'm finding it a lot of fun, and learning a lot!"

When she pulls these stunts in front of others, you will look infinitely classier than she for refusing to play the game.

The previous answers are quite good, and I'd like to add another possible line of defence.

If possible, minimise interaction with her. You will not learn anything from her (except how not to behave), and she is very unlikely to improve (miracles happen, but very rarely).

If you do not feel in the mood to give cheery/positive answers, another strategy is to respond blandly, such as with a politely interested "Really?"/"That sounds interesting"/"how lovely" or the like. Blandness and being boring (to her, not to others) takes the fun out of trying to upset you.

Most importantly, make clear to yourself that she has no role in your scientific life. She is not your boss or your adviser. You decide what you need. Don't watch how others in your department are doing. The only people you may need to watch are competitors in your research field at other institutions. The progress of some random colleague at your institution doesn't tell you anything about how well you are doing or should be doing. They may rush ahead, and fall back again with respect to you, you never know; fields are not comparable.

Your colleague clearly does not have the ability to judge with confidence where she is placed herself, and she is so worried about others that she tries to transfer this worry to you. Don't let that happen, you are not her emotional recycling bin. At this stage, however, which you describe I do not see a necessity yet to involve any third person.

To add a legal perspective (as a department chair, I received training regarding such matters), some of the behaviors you described will in my opinion very probably qualify as a form of workplace harassment in the United States, assuming that you can be considered an employee (which is not certain and would depend on the particulars of your status in the graduate program). The occurrence of such harassment could be argued to constitute a hostile work environment. If you complain to your advisor/PI/department chair, it will be their legal responsibility to take action to prevent the harassment. As the Wikipedia article I linked to above explains:

An employer can be held liable for failing to prevent these workplace conditions, unless it can prove that it attempted to prevent the harassment and that the employee failed to take advantage of existing harassment counter-measures or tools provided by the employer.

Even if you are not considered an employee, there may be other laws that apply to the situation, and regardless, common sense would dictate that your advisor and department are in a good position, and have good incentives, to address the problem should you choose to complain to them.

To conclude, I'm not saying that you necessarily should complain, but this is one option you should be aware of. Disclaimer: as I said I received some training on workplace harassment, but I am not a lawyer and would recommend that you seek more authoritative information on any legal-related issues before making any sensitive decisions.