How do I add validation to the form in my React component?

You should avoid using refs, you can do it with onChange function.

On every change, update the state for the changed field.

Then you can easily check if that field is empty or whatever else you want.

You could do something as follows :

class Test extends React.Component {
  constructor(props) {
    super(props);

    this.state = {
      fields: {},
      errors: {},
    };
  }

  handleValidation() {
    let fields = this.state.fields;
    let errors = {};
    let formIsValid = true;

    //Name
    if (!fields["name"]) {
      formIsValid = false;
      errors["name"] = "Cannot be empty";
    }

    if (typeof fields["name"] !== "undefined") {
      if (!fields["name"].match(/^[a-zA-Z]+$/)) {
        formIsValid = false;
        errors["name"] = "Only letters";
      }
    }

    //Email
    if (!fields["email"]) {
      formIsValid = false;
      errors["email"] = "Cannot be empty";
    }

    if (typeof fields["email"] !== "undefined") {
      let lastAtPos = fields["email"].lastIndexOf("@");
      let lastDotPos = fields["email"].lastIndexOf(".");

      if (
        !(
          lastAtPos < lastDotPos &&
          lastAtPos > 0 &&
          fields["email"].indexOf("@@") == -1 &&
          lastDotPos > 2 &&
          fields["email"].length - lastDotPos > 2
        )
      ) {
        formIsValid = false;
        errors["email"] = "Email is not valid";
      }
    }

    this.setState({ errors: errors });
    return formIsValid;
  }

  contactSubmit(e) {
    e.preventDefault();

    if (this.handleValidation()) {
      alert("Form submitted");
    } else {
      alert("Form has errors.");
    }
  }

  handleChange(field, e) {
    let fields = this.state.fields;
    fields[field] = e.target.value;
    this.setState({ fields });
  }

  render() {
    return (
      <div>
        <form
          name="contactform"
          className="contactform"
          onSubmit={this.contactSubmit.bind(this)}
        >
          <div className="col-md-6">
            <fieldset>
              <input
                ref="name"
                type="text"
                size="30"
                placeholder="Name"
                onChange={this.handleChange.bind(this, "name")}
                value={this.state.fields["name"]}
              />
              <span style={{ color: "red" }}>{this.state.errors["name"]}</span>
              <br />
              <input
                refs="email"
                type="text"
                size="30"
                placeholder="Email"
                onChange={this.handleChange.bind(this, "email")}
                value={this.state.fields["email"]}
              />
              <span style={{ color: "red" }}>{this.state.errors["email"]}</span>
              <br />
              <input
                refs="phone"
                type="text"
                size="30"
                placeholder="Phone"
                onChange={this.handleChange.bind(this, "phone")}
                value={this.state.fields["phone"]}
              />
              <br />
              <input
                refs="address"
                type="text"
                size="30"
                placeholder="Address"
                onChange={this.handleChange.bind(this, "address")}
                value={this.state.fields["address"]}
              />
              <br />
            </fieldset>
          </div>
        </form>
      </div>
    );
  }
}

React.render(<Test />, document.getElementById("container"));
    

In this example I did the validation only for email and name, but you have an idea how to do it. For the rest you can do it self.

There is maybe a better way, but you will get the idea.

Here is fiddle.