What should I do if my employer wants to pay cash without any records?

You should not do this, and should insist on proper withholding and that proper records be maintained. The employer's tax liability, if it is trying to play games with its tax obligations, is its conern--but you have a legitimate interest in all the following:

1) Making sure that the employer portion of FICA is paid, so that you are not held liable for it (i.e. if the employer doesn't pay its portion, you would owe another 6% plus in taxes)--you might also be liable for interest and/or penalties for not properly withholding;

2) Making sure that your earnings are property credited for purposes of accruing eligibility for Social Security;

3) Making sure there are payments made to the unemployment insurance system on your behalf, so that if fired, you have access to unemployment compensation;

4) Making sure there is a record of your employment for purposes of you having access, in the event of medical need or injury, to Worker's Compensation, disability, and FMLA leave, as appropriate.

You could be substantially injured by your employer's failure to keep records and account for taxes and other payments it owes on your behalf.

If your employer desires to pay you cash without any records, I suggest that you ask him or her to simply provide your with a receipt for the cash paid dated and signed by himself/herself with the amount paid. Explain to the employer that he will need documentation showing that you have been paid as well as you.

It makes no sense for an employer not to pay an employee by check since wages are deductible for a company with respect to taxes.