What allows the House of Representatives to fine its members and is there precedent for doing so?

Question

Speaker Nancy Pelosi and some House members are proposing a fine for members who bypass the security protocols of the House or fail to wear a mask on the floor.

Lawmakers will be fined $500 on a first offense and $2,500 for a second offense, and fines will be deducted from their pay, according to the aide. Fines cannot be paid with members' campaign funds or their expense accounts.

What allows the House of Representatives to fine its members? In other words, what gives the House the authority to fine its members? Is there precedent for fining members?

Answer 1

Yes, they can do this as a disciplinary measure against any of its members. It is just a standard procedure to help keep discipline and order in the chambers.

https://www.everycrsreport.com/reports/RL31382.html

The House of Representatives—in the same manner as the United States Senate—is expressly authorized within the United States Constitution (Article I, Section 5, clause 2) to discipline or “punish” its own Members. This authority of the House to discipline a Member for “disorderly Behaviour” is in addition to any criminal or civil liability that a Member of the House may incur for particular misconduct, and is used not merely to punish an individual Member, but to protect the institutional integrity of the House of Representatives, its proceedings, and its reputation.

The House may discipline its Members without the necessity of Senate concurrence. The most common forms of discipline in the House are now “expulsion,” “censure,” or “reprimand,” although the House may also discipline its Members in other ways, including fine or monetary restitution, loss of seniority, and suspension or loss of certain privileges. In addition to such sanctions imposed by the full House of Representatives, the standing committee in the House which deals with ethics and official conduct matters, the House Committee on Ethics—formerly called the Committee on Standards of Official Conduct—is authorized by House Rules to issue a formal “Letter of Reproval” for misconduct which does not rise to the level of consideration or sanction by the entire House of Representatives. Additionally, the Committee on Ethics has also expressed its disapproval of certain conduct in informal letters and communications to Members.

Answer 2

The House of Representatives has the constitutional authority to impose disciplinary measures on its members, including "expulsion, censure or reprimand, fines or other economic sanctions, and deprivation of seniority or committee status".

Article I, Section 5, Clause 2 of the US Constitution states:

Each House may determine the Rules of its Proceedings, punish its Members for disorderly Behaviour, and, with the Concurrence of two thirds, expel a Member.

There are some precedents listed in the House Practice: A Guide to the Rules, Precedents and Procedures of the House.

Pursuant to its constitutional authority to punish its Members (U.S. Const. art. I Sec. 5 clause 2), the House may levy a fine as a disciplinary measure against a Member for certain misconduct. Deschler Ch 12 Sec. 17. The fine may be coupled with certain other disciplinary measures deemed appropriate by the House. Thus, in one instance, the House disciplined a Member (for improper expenditure of House funds for private purposes) by imposing a fine of $25,000, to be deducted on a monthly basis from his salary. 91-1, Jan. 3, 1969, pp 29, 34. In another instance, in the 96th Congress, a Member was required to make restitution of monies in the amount which he had personally benefited in his misuse of the congressional clerk-hire allowance. 96-1, July 31, 1979, pp 21584-92. Fines imposed by the House are separate and distinct from those for which a Member might be liable under federal law.

(emphasis mine)

More precedents are listed in a CRS report (first linked by @JoeW in his answer):

The precedents in the House have demonstrated that the House fined a Member in 1969 the sum of $25,000 to be repaid by automatically withdrawing a certain amount regularly from his pay, for various conduct offenses, including the misuse of official committee appropriations, payroll, and expenses.⁷⁹ A Member of the House who was censured in 1979 was required to "make restitution of substantial amounts by which he was unjustly enriched," that is, the Member was expressly ordered within the resolution of censure to pay to the House a specific amount by executing an interest-bearing demand promissory note for $40,031.66, made payable to the Treasury of the United States.⁸⁰ A Member of the House who was officially "reprimanded" by the House in 2012 for misuse of official resources in compelling official congressional staff to work on political campaigns was also "fined" $10,000 as part of the reprimand.⁸¹ The Ethics Committee issued a "Letter of Reproval" to a Member in June 2014 regarding improper gifts and misuse of campaign funds, and the committee directed the Member to "repay the full amount of the improper gifts and the improperly used campaign funds" in the amount of $59,063.⁸²

The same CRS report, however, also noted that it was relatively rare to fine members for disciplinary purposes:

Fines for disciplinary purposes in the House, as well as in the Senate,⁷⁷ have been relatively infrequent occurrences.⁷⁸

[ ... ]

⁷⁸Studies have noted that prior to 1969, no Members of the House had ever been fined for disciplinary reasons. McLaughlin, “Congressional Self-Discipline: The Power to Expel, Exclude and to Punish,” 41 FORDHAM L.R. 43, 61 (Oct. 1972). There had in the 1800’s been a few instances noted in precedents where the House authorized fines for absences, or as a condition for discharge. Note, IV HINDS’ PRECEDENTS, supra at §§3011-3014.