Minimum Deposit
Minimum deposits vary from bank to bank, with most brick-and-mortar banks requiring as little as $100 (some banks might even have $0 deposit requirements). Check with your bank what would be their minimum deposit requirement for foreign clients.
Letter from a CPA or Company Attorney
This document might be required in some banks, if the bank is located in a state other than the state of registration. For example, if you register a Wyoming LLC and want to open an account in a New York bank, the bank would want to see a statement, written by a company accountant or an attorney, stating that this company does not do business in their respective state (in our example, New York). This should not be a complicated letter, something along the lines of, e.g., "I am such and such, confirming that company ABC, formed under the laws of Wyoming, does not and has no immediate plans to conduct business in the state of New York...". The letter must be dated and signed.
Certificates of Good Standing
Many banks require that companies submit Certificates of Good Standing (also called Certificate of Existence) to show that they are currently doing business and in good standing in the state in which they formed their business. Please contact the bank to see if they have this requirement.
Certificate of Incumbency
Certificate of Incumbency (also called "Incumbency Certificate") is an official document that lists the names of incumbent directors and officers within an organization, and their corporate position within it. An Incumbency Certificate is used as confirmation of the identity of the signing authorities of a company and to prove that they are authorized to enter into legally- binding transactions on the company's behalf. This certificate can be drafted on demand (there are plenty of samples online), and must be signed by the company secretary and sealed by a company seal.
Banking Resolution
Banking Resolution is a company document issued by the Board of Directors (for corporations) or by Members (for LLCs), giving certain individuals the authority to open a bank account on behalf of the company. Some banks have specific language that they require to include in the resolution (and would often supply a sample), other bank accept resolutions issued by the company, as long as it clearly presents and identity of the individual(s), and the extent of the authorization.