In a question like this, it becomes difficult to state one definitive answer at the cost of others. All of the senses are essential in being able to convince someone of anything. If the purpose of convincing is a valid one, something that an individual feels compelled to ensure that another accepts, then appealing to all of the senses becomes essential. The reason why it is essential to reach an individual through all of the senses is that it increases the propensity for an inroad to persuasion to develop. Advertisers understand this. Advertisers recognize that the appeal to all of the senses increases the chance for the consumer to be impacted by the advertisement. The reach of the advertiser increases when all of the senses are sought. To eliminate other senses decreases the ability to reach the consumer. It reduces possibility for connection and convincing as opposed to increasing it.
Part of this might reside in how different human beings are. For one person, the sense of touch is definitive in how they perceive reality, while for another, the sense of sight is critical in how they perceive reality. The method of perception varies from individual to individual. In the desire to convince someone of anything, appealing to all of the sense ensures that their particular style of perception is reached. In trying to convince an individual, being able to reach as many sensory level of perception as impossible creates an "in" with the target audience. It also helps to substantiate strength in convincing. If an individual is reached through their own dominant style as well as through secondary methods of perception, the target audience feels that the persuasion is stronger. Greater appeal emerges when more than one sensory level of perception is reached. It is for this reason that the appeal to as many senses as possible is critical in convincing an individual of a particular purpose or idea.
Like most of the people here, I would also say that the sense of sight is the most important of the five senses to appeal to when trying to convince someone of something. While it is a matter of appearing trustworthy in facial cues (eye contact, etc.) as well as body language, it is also important to consider how sight is important due to how visual people are as a whole. According to the links below, "presenters who use visual aids are 43% more effective in persuading audience members to take a desired course of action than presenters who don't use visuals," and viewers are "85% more likely to purchase a product after watching a product video." Sight is often the primary decider in how we interpret a situation. The links below provide another good example; if you tried to tell someone (sense of hearing) to pet your dog because he's super friendly, and that person looked down and saw a snarling, teeth-baring monstrosity at the end of your leash, you would have a far harder time convincing that person of the truth of your words. Humans are visual creatures, so if you want someone to believe you or be moved by your message, appeal to their eyes in some way.
In my opinion, the sense of trust is the most important thing while convincing someone. People will be convinced if you are trustworthy, if your words and acts offer a sense of credibility. It takes a long time to build up trust among two persons and once build, it will withstands all ups and down. It does not mean that the other person has to agree with everything that you say, they may have their own opinions and views. But it is important that the person believes in what you are saying.
Our society is set up in such a way that free-thinking persons are becoming rare nowadays. Often times the position or contacts of a person influences us whether we will be convinced by him or not. This may or may not be related to the person's accomplishment and potentials. It is truly unfortunate, but this is how it is.
I made a typo: withstand, not withstands!
In reference to the five basic senses, the sense of sight, touch, taste, smell and hearing; i think the most influential in convincing someone would be the sense of sight. As cliche as it sounds, the famous line "seeing is believing." might be applicable in this case.
For example, when buying something, the salesman would often times show you the product and it's amazing features. He would show you how it works, and how to use it. And when you see how cool the product is, you are eventually convinced to buy it.
Another example. When a friend talks to you about another person, and that person is described as scary, strict and stern. You immediately form a picture in your mind based on what you've heard. Then at one point, you meet this man in person and you see a harmless-looking, smiling old lady. Everything you've heard will now contradict what you see. You would now be confused and won't know whether this lady is scary or not.
The sense of sight plays a major role in convincing someone since majority of the information we take in comes visually.
Sense of sight. Body language makes up 55% of the message you convey, so what the other person sees of you is important when you are trying to convince. Assuming that the content of the speech is indeed persuasive, the sense of sight is important as the person observes your facial expression and your delivery.
If you are referring to our five senses then I would say the sense of sight. If you are going to convince someone of something, they will have to trust you and know everything is ok. The best way to trust someone is to see it by their actions, not just what they say. So, if you can see how someone acts and know that they are a trustworthy person, it will be easier to convince you.
Personally, the sense of sight is very important in deciding whether someone is trustworthy. If someone is trying to convince you, you look at the way the speak and their body movement. If someone is not being very truthful you can definitely see it in their body movement which is why the sense of sight is the most important in trying to convince someone.