Some people choose to quit their job to care for a family member that does not live in the same area as the employee. All avenues must be looked at prior to quitting because often the employer can make many different accommodations or provide an extended unpaid leave of absence to the employee. You should speak to your employer to find out if you are able to qualify for FMLA which can provide you leave while holding your job. When that is not possible once someone decides to quit it is important you notify the employer of exactly why you are quitting. In order to qualify for benefits in this case, the person you must be the only person that is able to care for the person. Typically they must basically be completely disabled or unable to care for themselves such as not being able to drive.
The goal with unemployment is to establish good cause for quitting. It is important to document your attempt to continue working with your employer prior to quitting. Also be prepared to describe how you are required to care for the person. If you were required to care for someone for only a short period of time, such as a week or so, it will be harder to qualify for benefits because many employers can be persuaded to provide leave for that amount of time. Quitting with good cause is more likely if you have to care for someone long term as in a few months or more.