First, using std::cin >> name will fail if the user enters John Smith because >> splits input on whitespace characters. You should use std::getline() to get the name:

std::getline(std::cin, name);

Here we go…

There are a number of ways to check that a string contains only alphabetic characters. The simplest is probably s.find_first_not_of(t), which returns the index of the first character in s that is not in t:

bool contains_non_alpha
    = name.find_first_not_of("abcdefghijklmnopqrstuvwxyz") != std::string::npos;

That rapidly becomes cumbersome, however. To also match uppercase alphabetic characters, you’d have to add 26 more characters to that string! Instead, you may want to use a combination of find_if from the <algorithm> header and std::isalpha from <cctype>:

#include <algorithm>
#include <cctype>

struct non_alpha {
    bool operator()(char c) {
        return !std::isalpha(c);
    }
};

bool contains_non_alpha
    = std::find_if(name.begin(), name.end(), non_alpha()) != name.end();

find_if searches a range for a value that matches a predicate, in this case a functor non_alpha that returns whether its argument is a non-alphabetic character. If find_if(name.begin(), name.end(), ...) returns name.end(), then no match was found.

But there’s more!

To do this as a one-liner, you can use the adaptors from the <functional> header:

#include <algorithm>
#include <cctype>
#include <functional>

bool contains_non_alpha
    = std::find_if(name.begin(), name.end(),
                   std::not1(std::ptr_fun((int(*)(int))std::isalpha))) != name.end();

The std::not1 produces a function object that returns the logical inverse of its input; by supplying a pointer to a function with std::ptr_fun(...), we can tell std::not1 to produce the logical inverse of std::isalpha. The cast (int(*)(int)) is there to select the overload of std::isalpha which takes an int (treated as a character) and returns an int (treated as a Boolean).

Or, if you can use a C++11 compiler, using a lambda cleans this up a lot:

#include <cctype>

bool contains_non_alpha
    = std::find_if(name.begin(), name.end(),
                   [](char c) { return !std::isalpha(c); }) != name.end();

[](char c) -> bool { ... } denotes a function that accepts a character and returns a bool. In our case we can omit the -> bool return type because the function body consists of only a return statement. This works just the same as the previous examples, except that the function object can be specified much more succinctly.

And (almost) finally…

In C++11 you can also use a regular expression to perform the match:

#include <regex>

bool contains_non_alpha
    = !std::regex_match(name, std::regex("^[A-Za-z]+$"));

But of course…

None of these solutions addresses the issue of locale or character encoding! For a locale-independent version of isalpha(), you’d need to use the C++ header <locale>:

#include <locale>

bool isalpha(char c) {
    std::locale locale; // Default locale.
    return std::use_facet<std::ctype<char> >(locale).is(std::ctype<char>::alpha, c);
}

Ideally we would use char32_t, but ctype doesn’t seem to be able to classify it, so we’re stuck with char. Lucky for us we can dance around the issue of locale entirely, because you’re probably only interested in English letters. There’s a handy header-only library called UTF8-CPP that will let us do what we need to do in a more encoding-safe way. First we define our version of isalpha() that uses UTF-32 code points:

bool isalpha(uint32_t c) {
    return (c >= 0x0041 && c <= 0x005A)
        || (c >= 0x0061 && c <= 0x007A);
}

Then we can use the utf8::iterator adaptor to adapt the basic_string::iterator from octets into UTF-32 code points:

#include <utf8.h>

bool contains_non_alpha
    = std::find_if(utf8::iterator(name.begin(), name.begin(), name.end()),
                   utf8::iterator(name.end(), name.begin(), name.end()),
                   [](uint32_t c) { return !isalpha(c); }) != name.end();

For slightly better performance at the cost of safety, you can use utf8::unchecked::iterator:

#include <utf8.h>

bool contains_non_alpha
    = std::find_if(utf8::unchecked::iterator(name.begin()),
                   utf8::unchecked::iterator(name.end()),
                   [](uint32_t c) { return !isalpha(c); }) != name.end();

This will fail on some invalid input.

Using UTF8-CPP in this way assumes that the host encoding is UTF-8, or a compatible encoding such as ASCII. In theory this is still an imperfect solution, but in practice it will work on the vast majority of platforms.

I hope this answer is finally complete!