Does the state of NY require that you withhold from the employee a portion for their share of SUI?

Yes; for 2009;employer's portion is: 1.3%---9.7% on first $8,500 with voluntary contributions. Employee's portion is: 0.5% of weekly wages, not to exceed $.60 per week( disability insurance).If an employee's weekly wage is, say, $600, then $600*0.005%=$2.50; $2.50>$.60; the employee pays $.60 per week.