Can I somehow convert it to switch-case?
You can (see below), but it probably wouldn't be best practice. You can simplify it a bit though:
private getClientBankAuthStatus(bankAuth: BankAuthenticationDto, isExpired: boolean): string {
    if (!bankAuth) {
        return "No";
    }
    return isExpired ? "Expired" : "Yes";
}
But for completeness, the switch version might be:
private getClientBankAuthStatus(bankAuth: BankAuthenticationDto, isExpired: boolean): string {
    switch (`${bankAuth ? "auth" : ""}-${isExpired ? "expired" : "ok"}`) {
        case "auth-ok":
            return "Yes";
        case "auth-expired":
            return "Expired";
        default:
            return "No";
    }
}
...or there's Nina's version, but again, I wouldn't use switch here at all.
Side note: I'd use a different return type on the function. Either use an enum rather than string literals, or use a union of string literals types:
private getClientBankAuthStatus(
    bankAuth: BankAuthenticationDto,
    isExpired: boolean
): "Yes" | "No" | "Expired" {
   ^^^^^^^^^^^^^^^^^^^^^^^^