Do basic solutions also have \[{H^ + }\] ions? If yes, then why are these basic?

Hint: Basic solutions are aqueous solutions of bases. They contain more numbers of \[O{H^ - }\] ions than \[{H^ + }\] ions. They are characterized by \[pH\] values in the range of \[7 - 14\].

Complete answer:
When bases are dissolved in water leads to formation of basic solutions. A base is any substance which accepts protons from an acid according to the Bronstead-Lowry concept. Common examples are sodium hydroxide, potassium hydroxide, sodium carbonate and sodium bicarbonate etc.
The aqueous solution of the bases has both \[{H^ + }\] ion and\[O{H^ - }\]. But the concentration of the \[O{H^ - }\] ion is greater than the \[{H^ + }\] ion. So the answer to the question is yes, basic solutions also have \[{H^ + }\] ion.
They are basic as the concentration of \[{H^ + }\] ion is less than the concentration of \[O{H^ - }\] ion. As a result the pH value lies in the range of \[7 - 14\]. In other words higher the concentration of \[O{H^ - }\] ion lower is the value of \[pOH\].
\[pOH\] is a measure of the negative logarithm of hydroxide ion concentration. The excess amounts of \[O{H^ - }\] ion present in basic solution accounts for the basicity of the solution and has lower \[pOH\] value.

Note:
Acids and bases combine with each other through a process of neutralization to produce salts. Several theories have been proposed by various scientists to identify an acid and a base in a reaction. Acids are referred to as proton donor and electron acceptor and bases are referred to as substances which donate electrons and accept protons. Strong bases react with water completely producing hydroxide ions and weak bases react with water partially producing hydroxide ions.