Who pays the fees?
It's a filing fee. So whoever files, pays the fee. Of-course, you may want to be reimbursed for this if the arbitration concludes the onus is on the other party and not on your side. But that's part of the resolution, not the initial filing.
Do note, you suggest the filing fee would be 1,500 minimum. For a business-to-business problem, this would be correct. However, I'm assuming you're more interested in disputes between users and Stack Exchange. Users would be consumers, not a business (unless we're talking about Teams here, in that case you'd be correct).
The full text of the section you linked to (emphasis mine):
For two-party matters, the Filing Fee is $1,500. For matters involving three or more parties, the filing fee is $2,000. The entire Filing Fee must be paid in full to expedite the commencement of the proceedings. Thereafter, a Case Management Fee of 12% will be assessed against all Professional Fees, including time spent for hearings, pre- and post-hearing reading and research and award preparation. JAMS also charges a $1,500 filing fee for counterclaims.
For matters involving consumers, the consumer is only required to pay $250. See JAMS Policy on Consumer Arbitrations Pursuant to Pre-Dispute Clauses. For matters based on a clause or agreement that is required as a condition of employment, the employee is only required to pay $400. See JAMS Policy on Employment Arbitrations, Minimum Standards of Fairness.
A refund of $600 will be issued if the matter is withdrawn within five days of filing. After five days, the filing fee is non-refundable.
User versus SE: $250
Employee versus SE: $400
Business versus SE: $1,500
Multiple businesses versus SE: $2,000
For completeness sake: should SE feel that they have a dispute with you and they end up with filing, the filing fees are on them. Those will probably be $1,500, regardless of who they have a problem with.