How to store username and password to API in wordpress option DB?

While I agree with the previous answers, to answer the question you actually asked, what comes to mind is to use one of these constants for wp-config.php:

define('AUTH_KEY',        'redacted');
define('SECURE_AUTH_KEY', 'redacted');
define('LOGGED_IN_KEY',   'redacted');
define('NONCE_KEY',       'redacted');

They are meant to be unique across wordpress installations – and are about the only options for pre-existing keys to be found in wordpress. Alternate would be to add your own similar constant that is built by hashing one of them against the admin email address or similar – and then storing that in a hidden setting option — to protect against losing your key if someone accidentally modifies the keys after your plugin is installed. The danger is, that if they were not made unique on the initial install, but the admin / site owner decides to rectify the failure after the fact, they shouldn’t accidentally break your password encryption.

As for encryption / decryption functions – a quick Google search returns the following listing with code that appears to fit the bill: http://maxvergelli.wordpress.com/2010/02/17/easy-to-use-and-strong-encryption-decryption-php-functions/

function encrypt($input_string, $key){
    $iv_size = mcrypt_get_iv_size(MCRYPT_RIJNDAEL_256, MCRYPT_MODE_ECB);
    $iv = mcrypt_create_iv($iv_size, MCRYPT_RAND);
    $h_key = hash('sha256', $key, TRUE);
    return base64_encode(mcrypt_encrypt(MCRYPT_RIJNDAEL_256, $h_key, $input_string, MCRYPT_MODE_ECB, $iv));
}

function decrypt($encrypted_input_string, $key){
    $iv_size = mcrypt_get_iv_size(MCRYPT_RIJNDAEL_256, MCRYPT_MODE_ECB);
    $iv = mcrypt_create_iv($iv_size, MCRYPT_RAND);
    $h_key = hash('sha256', $key, TRUE);
    return trim(mcrypt_decrypt(MCRYPT_RIJNDAEL_256, $h_key, base64_decode($encrypted_input_string), MCRYPT_MODE_ECB, $iv));
}

Here’s some documentation of the AES encryption used here: http://www.chilkatsoft.com/p/php_aes.asp