What is the ruling on accepting money from one's sister if its source is her travel, which is Islamically unlawful? And is one absolved of responsibility by advising her about the unlawfulness of her travel without a mahram to foreign countries?

Traveling for study without a mahram to a country where fitan (trials/temptations) are abundant, and where there is intermingling of the sexes, is reprehensible. The woman who embarks on such a journey has erred and must repent, as must anyone who encouraged her to do so. The mahram must reside with the woman in a country where incidents and fitan are prevalent. There is no doubt that seeking knowledge is commendable, as long as it does not lead to something forbidden. There is no sin upon you as long as you have advised your sister and did not assist her in her travel. As for the money your sister gives you, there is no objection to accepting it, unless your refusal would deter her and lead to her rectification; in that case, refusal is obligatory, as it falls under the category of boycotting for the purpose of enjoining the good and forbidding the evil. In all cases, a condition for accepting her gifts is that her work must be permissible (in the Sharia).