Can I only target bottom border at JavaFX Styling?

Here is a quick and dirty example of how that can be done.

import javafx.application.Application;
import javafx.geometry.Pos;
import javafx.scene.Scene;
import javafx.scene.control.TextField;
import javafx.scene.layout.BorderPane;
import javafx.scene.layout.HBox;
import javafx.scene.layout.VBox;
import javafx.stage.Stage;

public class TextFieldStyleTest extends Application {

    @Override
    public void start(Stage primaryStage) throws Exception {
        BorderPane root = new BorderPane();

        VBox vBox = new VBox();
        vBox.setAlignment(Pos.CENTER);
        root.setCenter(vBox);

        HBox hBox = new HBox();
        hBox.setAlignment(Pos.CENTER);
        vBox.getChildren().add(hBox);

        TextField textField = new TextField("Hello World");
        textField.setAlignment(Pos.BASELINE_CENTER);
        hBox.getChildren().add(textField);

        textField.setStyle("-fx-border-color: red; -fx-border-width: 0 0 10 0;"); 


        Scene scene = new Scene(root, 800, 600);
        primaryStage.setScene(scene);
        primaryStage.show();        
    }

    public static void main(String[] args) {
        launch(args);
    }

}

class TextFieldStyleTestLauncher {public static void main(String[] args) {TextFieldStyleTest.main(args);}}

Yes, it is possible to give each side a different color. From the JavaFX CSS Reference Guide, for Region:

• CSS Property: -fx-border-color
• Values: <paint> | <paint> <paint> <paint> <paint> [ , [<paint> | <paint> <paint> <paint> <paint>] ]*
• Default: null
• Comments: A series of paint values or sets of four paint values, separated by commas. For each item in the series, if a single paint value is specified, then that paint is used as the border for all sides of the region; and if a set of four paints is specified, they are used for the top, right, bottom, and left borders of the region, in that order. If the border is not rectangular, only the first paint value in the set is used.

^{Note: The above is actually from one row of a table, but Stack Overflow doesn't give a way of formatting things in a table.}

Meaning you can target the bottom border only by using:

.text-field {
  -fx-border-color: transparent transparent red transparent;
}

The -fx-border-width CSS property (and really all the CSS properties dealing with the Region#background and Region#border properties) behaves the same way. This means you can accomplish the same thing by setting the width of every side but the bottom to zero, just like in mipa's answer.

Here's an exaple using inline CSS (i.e. setStyle):

import javafx.application.Application;
import javafx.scene.Scene;
import javafx.scene.control.TextField;
import javafx.scene.layout.Region;
import javafx.scene.layout.StackPane;
import javafx.stage.Stage;

public class App extends Application {

  @Override
  public void start(Stage primaryStage) {
    TextField field = new TextField("Hello, World!");
    field.setStyle("-fx-border-color: transparent transparent red transparent;");
    field.setMaxWidth(Region.USE_PREF_SIZE);
    primaryStage.setScene(new Scene(new StackPane(field), 300, 150));
    primaryStage.show();

    // Remove blue outline from when TextField is focused. This
    // makes it easier to see the red border.
    primaryStage.getScene().getRoot().requestFocus();
  }
}

Which gives the following output:

Note that most of the "borders" added by modena.css (the default user-agent style sheet in JavaFX 8+) are not actually borders. Instead, they're multiple backgrounds with different insets.