Section 29-A of Article 2-B of the Executive section of the New York Consolidated Laws:
...The governor, by executive order, may issue any directive during a state disaster emergency declared in the following instances: ... epidemic, disease outbreak ... Any such directive must be necessary to cope with the disaster and may provide for procedures reasonably necessary to enforce such directive.
So, he's clearly got the authority to unilaterally issue directives in the event of an epidemic or disease outbreak, and provide mechanisms for enforcing those directives.
The relevant declaration of emergency is Executive Order 202. It doesn't contain any directives, just suspensions of certain laws. The fine comes from Executive Order 202.14, which modifies prior orders:
The enforcement of any violation of the foregoing directives [closure of businesses and prohibition on gatherings] on and after April 7, 2020, in addition to any other enforcement mechanism stated in any prior executive orders, shall be a violation punishable as a violation of public health law section 12-b(2) and the Commissioner of Health is directed and authorized to issue emergency regulations. The fine for such violation by an individual who is participating in any gathering which violates the terms of the orders or is failing to abide by social distancing restrictions in effect in any place which is not their home shall not exceed $1,000.
The prohibition on gatherings (the "social distancing" order) comes from Executive Order 202.10, which does not provide any penalty. The first penalty comes from Executive Order 202.11, which permits closing buildings in which gatherings take place as if they had exceeded the legal occupancy limits. Executive Order 202.14 is the first one to put monetary fines on gatherings.
The first executive order that provides a penalty for violating the business-closure orders is 202.8:
The provisions of Executive Order 202.6 [reduction of in-person workforce to 50%] are hereby modified to read as follows: Effective on March 22 at 8 p.m. ... Any business violating the above order shall be subject to enforcement as if this were a violation of an order pursuant to section 12 of the Public Health Law.
Section 12 of the Public Health Law:
1. (a) Any person who violates, disobeys or disregards any term or provision of this chapter or of any lawful notice, order or regulation pursuant thereto for which a civil penalty is not otherwise expressly prescribed by law, shall be liable to the people of the state for a civil penalty of not to exceed two thousand dollars for every such violation.
(b) The penalty provided for in paragraph (a) of this subdivision may be increased to an amount not to exceed five thousand dollars for a subsequent violation...
So in summary, the "raised the fine" statement is wrong. Prior to April 7, as far as I can tell, there was no monetary penalty for individuals violating the order. For businesses violating the closure and work-at-home orders, the fine is unchanged, at a maximum of $2000 for the first violation and $5000 for subsequent violations.
Only tangentially related, but an emergency declaration isn't an unrestricted grant of power. Section 29-A provides the following limitations:
b. no suspension or directive shall be made which is not in the interest of the health or welfare of the public and which is not reasonably necessary to aid the disaster effort;
e. any such suspension order or directive shall provide for the minimum deviation from the requirements of the statute, local law, ordinance, order, rule or regulation suspended consistent with the goals of the disaster action deemed necessary;
4. The legislature may terminate by concurrent resolution executive orders issued under this section at any time.
Any directive or suspension can be overridden by the courts if it goes beyond what is necessary, and can be overridden by the legislature for any reason.