From the southeastern United States where I am, $13 an hour is a darn fine wage. Many places have been cutting pay lately. Thank goodness we have minimum wage laws still in place.
That his salary has been cut from $26 an hour may be a terrible shock to the OP, but really, the amount he could draw in unemployment benefits, even if he qualifies for the maximum in NY state, is going to be less than his weekly wage while working even at $13 an hour. In other words, Sunday through Saturday, gross wages, if he gets $13 an hour for 40 hours of work, that's $520.00 grossed in a week. This is more than the NY maximum benefit by over $100 So no, partial unemployment insurance isn't going to be an option for this guy after the first of the year.
In order to receive partial benefits, you have to have worked and made less in gross wages than your weekly benefit amount would have been. This is not going to be an option for you. They don't consider how much it costs you to work or what your commuting expenses were. This is gross wages only, for the hours they have available for you. If it's more than your weekly benefit amount would be, then you cannot get partial unemployment.
If he only worked say two days in that week, and made $208 in gross wages, then he could file for partial unemployment. They'd allow some and then pay a suppliment up to the amount he could draw in weekly unempoyment benefits. But if he works full 40 hour weeks, even $13 an hour is enough that he'd not qualify for partial benefits.
To quit the job because his wages have been slashed in half and be approved for benefits,....it is possible to be approved under these circumstances. Not a shoo in, but quite possible. The employer has dramatically changed the basic hiring agreement.
But as so many people have pointed out, you'd be making less in unemployment benefits than you'd make working, are you sure you want to go this route while looking for that next job? If you're positive that there is a job out there that you can get that pays much more than $13 an hour, that the job is not available for you to go to now, but will be there after the first of the year, that it is just too costly for you to continue working at the old job at the reduced salary after it goes into effect....even if you're not sure you'll be approved for unemployment benefits, not sure you'll find that new job soon...then perhaps you'll choose to quit the job. As I said, there's a fair chance you can be approved for benefits under these circumstances.
If this is your choice sure that you wait until the last minute, until you have drawn out everything you can possibly get from them at the current pay rate. You don't want your employer to say you quit because of an unjustified rumor, or that they weren't really going to do that, but you just quit anyhow. Keep all the paperwork, any explanation of plans they've put out, email or paperwork regarding the planned pay reduction.
Then, and this is VERY IMPORTANT, don't work at all for the new reduced salary. Formally leave the job, be very plain with them about why you are resigning. DO NOT work for any length of time at the new reduced rate. To do so would be acceptance of the new pay terms, and will make it much harder to get approved for unemployment benefits.
Then file a claim for benefits in the first week you are not working. Explain to them, and present paper documentation if you can get it, that your employer cut your salary in half to encourage you to leave and avoid paying unemployment insurance benefits. You will have to wait several weeks for the unemployment insurance system to process this and approve or deny your claim, with no income during that time. You'll have to register for work and make many worksearches during the weeks you're drawing. If you don't find anything soon, you'll find yourself having to consider jobs that do pay quite a bit less than you were making.
If you really think you can do better soon, you'll have to consider all these options and make your decision. Remember, once you've worked at the reduced wage, there's no changing your mind and quitting and being approved to draw unemployment.
If you don't have a union contract, there's really nothing legal to prohibit an employer from slashing wages like this,but honestly, I hope you can soon find another job and move on. As someone said, they've practically announced to the housetops that they don't value you and don't give a care whether they have a dedicated and motivated workforce or not.