There is nothing wrong with a Muslim working in the field of website design, which is permissible in and of itself, but he has to avoid designing websites that promote haraam actions, such as websites for banks, alcohol, movies, chat between males and females, and so on.
The basic principle in forbidding that which is prohibited is the verse in which Allah, may He be exalted, says (interpretation of the meaning): “but do not help one another in sin and transgression. And fear Allah. Verily, Allah is Severe in punishment” [al-Maa’idah 5:2].
If the website is not solely for something haraam, and is a site for something that is basically permissible, but will include some haraam things -- as in the case of the company you asked about -- then it is permissible for you to design a website for them, subject to two conditions:
1. That most of what this company does is permissible; it should not be the case that most of their work or activity is in haraam fields.
2. That you yourself do not work on the haraam areas.
With regard to the company that you asked about: it is permissible for you to design a website for them, so long as the riba-based transactions mentioned are not the main activity of this company and that you do not make a window for this haraam activity or promote it in any way.
With regard to your telling your cousin about the Islamic ruling on what you abstain from, this is a kind of calling to Islam (da‘wah). Rather it is your working to design websites for haraam services that is bad for Islam. There is no justification for a Muslim to do something haraam in order to please a kaafir. In fact the kaafir himself may think that you are contradicting yourself if he knows that your religion forbids riba but you are designing a webpage for services that help people to get riba-based loans!
The Muslim has to beware of seeking people’s approval by means of that which angers Allah, may He be exalted, because that will backfire on him, leading to negative consequences and harm. At-Tirmidhi (2414) narrated (in a report that was classed as saheeh by al-Albaani in Saheeh at-Tirmidhi) that ‘Aa’ishah (may Allah be pleased with her) said: I heard the Messenger of Allah (blessings and peace of Allah be upon him) say: “Whoever seeks to please Allah by angering the people, Allah will suffice him against the people, but whoever seeks to please people by angering Allah, Allah will leave him to the people.”
There is another report, narrated by Ibn Hibbaan in his Saheeh (1/501): “Whoever seeks to please Allah by angering people, Allah, may He be exalted, will be pleased with him and will cause the people to be pleased with him; whoever seeks to please people by angering Allah, Allah will be angry with him and will cause people to be angry with him.”
It should be noted that true da‘wah and attracting people to Islam is done by adhering to it. But if you tell people that Islam forbids riba -- for example -- then you deal in it or help others with it, this is a kind of blocking the way of Allah and putting people off His religion, because they will say: If what he is calling us to and telling us about is true, why is he not adhering to it?
And Allah knows best.