1. Let $f(x)=x^7+5x^3+3x+1.$ We need the number of real solutions of $f(x)=0.$

2. Check monotonicity using derivative: $f'(x)=7x^6+15x^2+3.$

Now,

• $7x^6\ge 0$ for all real $x$
• $15x^2\ge 0$ for all real $x$
• $3>0$

Hence, $f'(x)=7x^6+15x^2+3>0 \quad \text{for all } x\in\mathbb R.$

So $f(x)$ is strictly increasing on $\mathbb R$.

3. A strictly increasing continuous function can cross the $x$-axis at most one point.

4. Now check whether it crosses the axis at least once. Since $f(x)$ is a polynomial, it is continuous. Also, $\lim_{x\to -\infty} f(x)=-\infty,$ because the leading term is $x^7$, and $\lim_{x\to +\infty} f(x)=+\infty.$

Therefore, by the Intermediate Value Theorem, $f(x)=0$ has at least one real root.

5. Combining steps 3 and 4:

• at most one real root,
• at least one real root.

Hence, the equation has exactly one real solution.

6. Therefore, the correct option is $\boxed{\text{B: }1}.$