以下是一个使用Apache HttpClient进行基本身份验证和会话Cookie的示例代码：

import org.apache.http.HttpEntity;
import org.apache.http.HttpHost;
import org.apache.http.auth.AuthScope;
import org.apache.http.auth.UsernamePasswordCredentials;
import org.apache.http.client.CredentialsProvider;
import org.apache.http.client.methods.CloseableHttpResponse;
import org.apache.http.client.methods.HttpGet;
import org.apache.http.client.protocol.HttpClientContext;
import org.apache.http.impl.auth.BasicScheme;
import org.apache.http.impl.client.BasicAuthCache;
import org.apache.http.impl.client.BasicCredentialsProvider;
import org.apache.http.impl.client.CloseableHttpClient;
import org.apache.http.impl.client.HttpClients;
import org.apache.http.impl.cookie.BasicClientCookie;

import java.io.BufferedReader;
import java.io.IOException;
import java.io.InputStreamReader;
import java.net.URI;
import java.net.URISyntaxException;

public class HttpClientExample {
    public static void main(String[] args) throws URISyntaxException, IOException {
        // 设置代理主机和端口
        HttpHost proxy = new HttpHost("proxy.example.com", 8080);

        // 创建HttpClient实例
        CloseableHttpClient httpClient = HttpClients.createDefault();

        // 创建Basic身份验证凭据提供程序
        CredentialsProvider credentialsProvider = new BasicCredentialsProvider();
        credentialsProvider.setCredentials(
                new AuthScope(AuthScope.ANY),
                new UsernamePasswordCredentials("username", "password"));

        // 创建HTTP代理上下文
        HttpClientContext context = HttpClientContext.create();
        context.setCredentialsProvider(credentialsProvider);
        context.setAuthCache(new BasicAuthCache());
        context.getAuthCache().put(proxy, new BasicScheme());

        // 创建GET请求
        URI uri = new URI("http://example.com/api/resource");
        HttpGet httpGet = new HttpGet(uri);

        // 执行请求
        CloseableHttpResponse response = httpClient.execute(proxy, httpGet, context);

        try {
            // 处理响应
            HttpEntity entity = response.getEntity();
            BufferedReader reader = new BufferedReader(new InputStreamReader(entity.getContent()));
            String line;
            while ((line = reader.readLine()) != null) {
                System.out.println(line);
            }
        } finally {
            // 关闭响应和HttpClient
            response.close();
            httpClient.close();
        }
    }
}

在上面的示例代码中，我们首先定义了代理主机和端口。然后，我们创建了一个CloseableHttpClient实例，并设置了Basic身份验证凭据提供程序，使用提供的用户名和密码。

接下来，我们创建了一个HttpClientContext对象，并将凭据提供程序和BasicAuthCache设置到上下文中。然后，我们将代理主机和BasicScheme放入AuthCache中。

然后，我们创建了一个HttpGet对象，并使用代理和上下文执行了请求。最后，我们处理响应并关闭响应和CloseableHttpClient。

请注意，示例代码中的用户名、密码、代理和URL应该根据实际情况进行更改。