Under U.S. law, employers are required to cover certain mandatory fees for H-1B petitions; employees cannot be forced to pay required fees such as the base filing fees or ACWIA training fees. 

For optional fees, like premium processing, the employer or the employee may cover them; typically the party who requests the premium processing pays it

Regarding the new $100,000 fee: the law as of now indicates that this fee is the employer’s responsibility. There is no clear directive yet that employees can or will be asked to contribute; legal/historical precedent suggests required fees remain employer’s burden.