Can only one member open a checking account
Generally in a partnership, one member can bind the others and the each share equally in the profits and losses, and agency law would provide that 3rd parties would treat each partner as if they can bind the partnership. LLCs are structurally a lot like partnerships.
You should trust your partners/co-members, but you should also have a written LLC Operating Agreement that spells out what members can do and not do, to make sure everyone understands how the LLC will do business. You can organize the membership authority of your LLC any way you want, so if your LLC wants to operate with single member authority, you can do that, and if you want the LLC to only transact business with unanimous approval and enter into contracts with all the members signing, you can do that too.
I think usually banks require only a copy of the entity's articles of organization to open an account, and they won't inquire as to the inner workings of an entity. As mentioned, 3rd parties can assume that any one partner/member is authorized to act for the entity.
Your LLC entity needs to have a governing document that lays out the operation rules and sets limits on the members/partners.
Disclaimer: Please note that this answer does not constitute legal advice, and should not be relied on, since each state has different laws, each situation is fact specific, and it is impossible to evaluate a legal problem without a comprehensive consultation and review of all the facts and documents at issue. This answer does not create an attorney-client relationship.